Construction and classification of symmetry protected topological phases in interacting fermion systems

The classification and lattice model construction of symmetry protected topological (SPT) phases in interacting fermion systems are very interesting but challenging. In this paper, we give a systematic fixed point wavefunction construction of fermionic SPT (FSPT) states for generic fermionic symmetry group $G_f=\mathbb Z_f\rtimes G_b$ which is a central extension of bosonic symmetry group $G_b$ (may contain time reversal symmetry) by the fermion parity symmetry group $\mathbb Z_f=\{1,P_f\}$. Our construction is based on the concept of equivalence class of finite depth fermionic symmetric local unitary (FSLU) transformations and decorating symmetry domain wall picture, subjected to certain obstructions. We will also discuss the systematical construction of boundary anomalous SPT (ASPT) states which leads to a trivialization of the corresponding bulk FSPT states. Thus, we conjecture that the obstruction free and trivialization free construction naturally leads to a classification of FSPT phases. Each fixed-point wave function admits an exactly solvable commuting-projector Hamiltonian. We believe that our classification scheme can be generalized to point/space group symmetry as well as continuum Lie group symmetry.

VI. Fixed-point wave function and classification of FSPT states in 3D 36 A. Four layers of degrees of freedom systems. Therefore, a systematical understanding for the classification and construction of SPT phases in interacting fermion systems is very desired.
Very recently, based on the concept of equivalence class of finite depth fermionic SLU (FSLU) transformation and decorated symmetry domain wall picture, a breakthrough has been made on the full construction and classification of FSPT states with a total symmetry G f = G b × Z f 2 (where G b is the bosonic unitary symmetry and Z f 2 is the fermion parity conservation symmetry) [53]. The fixed-point wave functions generated by FSLU transformations can be realized by exactly solvable lattice models and the resulting classification results all agree with previous studies in 1D and 2D using other methods [43,52,[54][55][56][57]. Most surprisingly, such a completely different physical approach precisely matches the potential global anomaly for interacting fermion systems classified by spin cobordism theory [10,[58][59][60][61][62][63][64].
It turns out that the mathematical objects that classify 1D FSPT phases with a total symmetry G f = G b × Z f 2 can be summarized as two cohomology groups of the symmetry group G b : H 1 (G b , Z 2 ) and H 2 (G b , U (1)), which correspond to the complex fermion decoration on G b symmetry domain walls and classification of 1D BSPT phases. The mathematical objects that classify 2D FSPT phases with a total symmetry G f = G b × Z f 2 are slightly complicated and can be summarized as three cohomology groups of the symmetry group G b [55,59]: H 1 (G b , Z 2 ), BH 2 (G b , Z 2 ), and H 3 (G b , U (1)). H 1 (G b , Z 2 ) corresponds to the Majorana chain decoration on G b symmetry domain walls. Naively, one may expect that the complex fermion decorations on the intersection point of G b symmetry domain walls should be described by the data H 2 (G b , Z 2 ). However, it turns out that such a decoration scheme will suffer from obstructions, and only the subgroup BH 2 (G b , Z 2 ) classifies valid and inequivalent 2D FSPT phases. More precisely, BH 2 (G b , Z 2 ) is defined by n 2 ∈ H 2 (G b , Z 2 ) that satisfy Sq 2 (n 2 ) = 0 in H 4 (G b , U (1)), where Sq 2 is the Steenrod square operation, Sq 2 : [65]. Finally, H 3 (G b , U (1)) is the well-known classification of BSPT phases.
Similarly, The mathematical objects that classify 3D FSPT phases with a total symmetry G f = G b × Z f 2 can also be summarized as three cohomology groups of the symmetry group G b [53,62]:BH 2 (G b , Z 2 ), BH 3 (G b , Z 2 ), and H 4 rigid (G b , U (1)). As a subgroup of H 2 (G b , Z 2 ),BH 2 (G b , Z 2 ) corresponds to the Majorana chain decoration on the intersection lines of G b symmetry domain walls subject to a much more subtle and complicated objections related to discrete spin structure. Again, as a subgroup of H 3 (G b , Z 2 ), BH 3 (G b , Z 2 ) corresponds to the complex fermion decoration on the intersection points of G b symmetry domain walls. And it is formed by elements n 3 ∈ H 3 (G b , Z 2 ) that satisfy Sq 2 (n 3 ) = 0 in H 4 (G b , U (1)). Finally, H 4 rigid (G b , U (1)) ≡ H 4 (G b , U (1))/Γ 4 corresponds to stable BSPT phases when embedded into interacting fermion systems. We note that Γ 4 is a normal subgroup of H 4 (G b , U (1)) generated by (−) Sq 2 (n2) , where n 2 ∈ H 2 (G b , Z 2 ) and (−) Sq 2 (n2) are viewed as elements of H 4 (G b , U (1)). Physically, Γ 4 corresponds to those trivialized BSPT phases when embedded into interacting fermion systems.
In this paper, we aim to generalize the above constructions and classifications of FSPT phases to generic fermionic symmetry group G f = Z f 2 × ω2 G b , which is a central extension of bosonic symmetry group G b (may contain time reversal symmetry) by the fermion parity symmetry group Z f 2 = {1, P f }. We will show that the equivalence class of finite depth FSLU transformation and decorated symmetry domain wall picture still apply for generic cases, subjected to much more complicated obstruction conditions. Moreover, we will also clarify the physical meaning of obstruction by introducing the notion of anomalous SPT (ASPT) states [66], that is, a new kind of SPT states that can only be realized on the boundary of certain SPT states in one dimension higher. In the meanwhile, this also implies that the corresponding bulk SPT states are actually trivialized. Finally, we will show that if G b is time reversal symmetry, an additional layer of p + ip topological superconducting state decoration on the symmetry domain wall will lead to new FSPT states, which is the analogy of decorating the E 8 state onto the symmetry domain wall for BSPT phases with time reversal symmetry [8].
B. Some generalities of fermionic symmetry groups For a fermionic system with total symmetry group G f , there is always a subgroup: the fermion parity symmetry group Z f 2 = {1, P f = (−1) F }, where F is the total fermion number operator. The subgroup Z f 2 is in the center of G f , because all physical symmetries should not change the fermion parity of the state, i.e., commute with P f . Therefore, we can construct a quotient group G b = G f /Z f 2 , which we will call bosonic symmetry group. Conversely, for a given bosonic symmetry group G b , there are many different fermionic symmetry group G f which is the central extension of G b by Z f 2 . We have the following short exact sequence: Different extensions G f are specified by 2-cocycles ω 2 ∈ H 2 (G b , Z 2 = {0, 1}). This is the reason why we denote G f as Z f 2 × ω2 G b . The group element g f of G f has the form g f = (P n(g) f , g b ) ∈ Z f 2 × G b , with n(g) = 0, 1. We may also simply denote it as g f = P n(g) f g b . And the multiplication rule in G f is given by where we have P n(g)+n(h)+ω2(g b ,h b ) f ∈ Z f 2 and g b h b ∈ G b . The associativity condition of g · h · k (g, h, k ∈ G b ) gives rise to the cocycle equation for ω 2 : (dω 2 )(g, h, k) := ω 2 (h, k) + ω 2 (gh, k) + ω 2 (g, hk) + ω 2 (g, h) = 0 (mod 2). ( We will omit the subscript of g b and use merely g to denote the group element of G b henceforth. One can also show that adding coboundaries to ω 2 will give rise to isomorphic G f . Therefore, ω 2 is an element in H 2 (G b , Z 2 ) and classifies the central extension of G b by Z f 2 . Note that there is another constraint for ω 2 as ω 2 (e, g) = ω 2 (g, e) = 0 (e is the identity element of G b ).
Another ingredient of the symmetry group is associated with time reversal symmetry which is antiunitary. We can use a function s 1 with s 1 (g) = 0, g is unitary 1, g is antiunitary (4) to indicate whether g ∈ G b is antiunitary or not. The function s 1 is a group homomorphism from G b to Z 2 because of the property (ds 1 )(g, h) := s 1 (h) + s 1 (gh) + s 1 (g) = 0 (mod 2).
So s 1 can also be viewed as a 1-cocycle in H 1 (G b , Z 2 ). Let us consider some examples. The superconductor with time reversal symmetry T 2 = P f (T 2 = −1 when acting on single fermion states) has bosonic symmetry group G b = Z T 2 = {e, T } and fermionic symmetry group In terms of our language, the 2-cocycle ω 2 and 1-cocycle s 1 have nonzero values ω 2 (T, T ) = 1 and s 1 (T ) = 1. They are nontrivial cocycles in H 2 (Z T 2 , Z 2 ) = Z 2 and H 1 (Z T 2 , Z 2 ) = Z 2 , respectively. By choosing different ω 2 and s 1 , we have three other fermionic symmetry groups G f : Z f 2 × Z 2 (trivial ω 2 and trivial s 1 ), Z f 4 = Z f 2 × ω2 Z 2 (nontrivial ω 2 and trivial s 1 ) and Z f 2 × Z T 2 (trivial ω 2 and nontrivial s 1 ). We will calculate the classifications of FSPT phases with these four fermionic symmetry groups in Appendix E 2.
C. Summary of main results

Summary of data and equations
data \ dim 1D 2D 3D phase factor n2 complex fermion n2 Kitaev chain phase factor TABLE I: Layers of classification data. The cochains n d ∈ C d (G b , Z 2 ), n d−1 ∈ C d−1 (G b , Z 2 ) and n d−2 ∈ C d−2 (G b , Z T ) describe the decorations of 0D complex fermions, 1D Kitaev chains, and 2D p + ip superconductors (SC) in the dspacial dimension model respectively. And ν d+1 ∈ C d+1 (G b , U (1) T ) is the bosonic U (1) phase factor in the wave function, which is related to the group cohomology classification of BSPT phases in d-spacial dimension. There is also an n 0 data in each dimension, if we want to classify fermionic invertible topological orders. We omit these n 0 states in this paper, for they do not need any G b symmetry protection. But they are important in the trivialization group Γ 2 for ν 2 or n 2 .
As discussed above, to specify the total symmetry group G f of a fermionic system, we have an 1-cocycles s 1 ∈ H 1 (G b , Z 2 ) which is related to time reversal symmetry and a 2-cocycle ω 2 ∈ H 2 (G b , Z 2 ) which tells us how G b is extended by Z f 2 . They satisfy the (mod 2) cocycle equations: Given the input information of the total symmetry group G f (i.e., G b with s 1 and ω 2 ), we summarize the classification data, symmetry conditions, consistency equations and extra coboundary (states trivialized by ASPT state in one lower dimensions) for FSPT states in different physical dimensions in Eqs. (8)- (19) (see also Table I for the  classification data, and Table II for the physical meanings of the consistency equations).
We note that the cochains n d ∈ C d (G b , Z 2 ), n d−1 ∈ C d−1 (G b , Z 2 ) and n d−2 ∈ C d−2 (G b , Z T ) describe the decorations of 0D complex fermions, 1D Kitaev chains, and 2D p + ip superconductors (SC) in the d-spacial dimension model respectively. In 1D, it is only possible to decorate complex fermion onto the G b symmetry domain wall and the constraint dn 1 = 0 is nothing but the fermion parity conservation requirement for a valid FSLU transformation. In 2D, it is possible to decorate both Majoran chain onto the G b symmetry domain wall and compelx fermion onto the intersection point of G b symmetry domain walls. In order to construct FSPT states, we must decorate closed Majoran chain onto the G b symmetry domain wall and this implies dn 1 = 0. Again, fermion parity conservation of FSLU transformation requires that dn 2 = (ω 2 + s 1 n 1 ) n 1 . In 3D, it is even possible to decorate 2D p + ip SC state onto the G b symmetry domain wall if G b contains anti-unitary symmetry. However, in order to construct such FSPT states, we must require that there is no chiral Majorana mode on the intersection lines of G b symmetry domain walls. Furthermore, dn 2 = (ω 2 + s 1 n 1 ) n 1 corresponds to the absence of free Majorana fermion on the intersection points of G b symmetry domain walls and dn 3 = (ω 2 + n 2 ) n 2 + s 1 (n 2 1 n 2 ) again corresponds to fermion parity conservation of FSLU transformation. Finally, the bosonic U (1)-valued phase factor ν d+1 ∈ C d+1 (G b , U (1) T ) must satisfy the so-called twisted cocycle condition dν i = O i+1 , which are generated by fixed point conditions of FSPT wave functions. We note that the bosonic layer data ν d+1 without superscript always means the inhomogeneous cochain in the twisted cocycle equation. The homogeneous cochain is obtained by a symmetry action and may have additional sign factors. There is also a symmetry action on the first term of the coboundary definition in dν d+1 . Because time reversal symmetry has nontrivial actions on both Z T and U (1) T , there is an exponent 1 − 2s 1 (g 0 ) = ±1 for the first term of dν d+1 .
Based on the above decoration construction, we can obtain the FSPT classifications by solving the consistency equations layer by layer as shown in Table II. The solutions of these equations can be used to construct FSPT states. And the final classifications are obtained from these data by quotient some subgroups. We note that B i are the coboundary subgroups defined for the corresponding cochain groups C i in the usual sense. The trivialization subgroups Γ i of the classification data correspond to the states that are trivialized by boundary ASPT states. In d spacial dimensions, the U (1) factor ν d+1 in Γ d+1 corresponds to BSPT state trivialized by fermions [52]. The complex fermion decoration data n d in the next layer Γ d is trivialized by boundary ASPT states with Kitaev chains [66]. And the Kitaev chain decoration data n d−1 in Γ d−1 is trivialized by boundary ASPT states with 2D p + ip chiral superconductors.

Summary of classification examples
Using the above data, we calculate the classifications for FSPT phases with several simple symmetry groups. They are summarized in Table III. Some of the derivations are given in Appendix E. In particular, we calculate the classifications for 2D FSPT phases with arbitrary unitary finite Abelian group G f in Appendix E 1. Our results are exactly the same as that in Ref. 13, which uses a totally different approach by investigating the braiding statistics of the gauge flux. The calculations for 3D FSPT phases with arbitrary unitary finite Abelian group G f are given in Ref. 68. The results are also consistent with 3D loop braiding statistics approaches. We calculate the classifications of FSPT phases for the four fermionic symmetry groups with G b ∼ = Z 2 in Appendix E 2. They are also consistent with previously known results. As an example of non-Abelian G f , we calculate the FSPT phases with quaternion group The rest of the paper is organized as follows. In section II, we review the key concept of SLU transformations. Using this approach, we show the classifications of BSPT phases in various dimensions. In section III, we summarize the procedures of constructing FSPT states. The definition of FSLU transformations is given in section III A. All layers of degrees of freedom and their symmetry transformation rules are summarized in section III B. In section III C, we discuss briefly the two essential requirements of the FSLU transformations: the coherence equations and the symmetry conditions. Using the outlined procedure, the details of the classifications of 1D, 2D and 3D FSPT phases are given in sections IV, V, and VI, respectively. In each dimension, we first give the symmetric decoration procedures. Then the F move (FSLU transformation) and its coherence equation are given explicitly. As the final step in classifying FSPT phases in each dimension, we discuss some new coboundaries associated with ASPT states in one lower dimension. We summarize this work in section VII.
In Appendix A, we show the classification of the simplest 0D FSPT phases. In Appendix B, we list all possible 2D and 3D moves that admit a branching structure. In Appendix C, the (local) Kasteleyn orientations for 2D and 3D lattices are discussed briefly. In Appendix D, we discuss the Bockstein homomorphism mapping a Z 2 -valued cocycle to a Z T -valued cocycle. It is useful in checking whether the obstruction function (−1) f k , where f k is a Z 2 -valued cocycle, is a U (1) T -valued coboundary or not. The detail calculations of FSPT phases for some simple groups are given in Appendix E. Some of the results are already summarized in Table III.

A. SLU transformation and BSPT phases
From the definition of SPT states, it is easy to see that (in the absence of global symmetry): Namely, an SPT state can be connected to a trivial state (e.g., a product state) vial LU transformation (in the absence of global symmetry). Clearly, Eq. (20) implies that the support space [69] of any SPT state in a region must be one dimensional. This is simply because a trivial state (e.g., a product state) has a one dimensional support space, and any SPT state will become a product state via a proper local basis change (induced by a LU transformation).
In the presence of global symmetry, we can further introduce the notion of symmetric local unitary (SLU) transformations classifying SPT phases in interacting bosonic systems. By SLU transformation, we mean the corresponding piecewise LU operator is invariant under symmetry G. More precisely, we have U pwl = i e −iH b (g i0 ,g i1 ,g i2 ,··· ) ≡ i U (g i0 , g i1 , g i2 , · · · ) and U (gg i0 , gg i2 , gg i3 , · · · ) = U (g i0 , g i1 , g i2 , · · · ) for any g ∈ G. (We note that here we choose the group element basis g i0 , g i1 , g i2 , · · · to represent bosonic symmetric unitary operator acting on a region labeled by i.) However, we need to enforce the SLU transformations to be one dimensional (when acting on the support space ρ A for any region A), and we call them invertible SLU transformations. Thus, we claim that SPT phases in interacting bosonic systems can be classified by equivalence class of invertible SLU transformations.
SPT phases are also referred as invertible (non-chiral) topological phases. It turns out that the novel concept of invertible SLU transformation even allows us to construct very general fixed-point SPT states. All of these fixedpoint wave functions admit exactly solvable parent Hamiltonians consisting of commuting projectors on an arbitrary triangulation with an arbitrary branching structure.

B. Fixed-point wave function and classification for BSPT phases in 1D
As a warm up, let us begin with fixed-point wave function in 1D and use SLU transformation to derive the well known classification results of 1D BSPT phases. Without loss of generality, here we assume that every (locally ordered) vertex i of the 1D lattice has bosonic degrees of freedom labeled by a group element g i ∈ G.
Our 1D fixed-point state is a superposition of those basis states with all possible 1D graph with a branching structure. |Ψ = all conf. Ψ g 0 g 1 g 2 g 3 g 4 g 5 g 0 g 1 g 2 g 3 g 4 g 5 .
In the following, we will derive the rules of wave function renormalization generated by SLU transformations for the above wave function and show how to construct all BSPT states in 1D. To obtain a fixed-point wave function, we need to understand the changes of the wave function under renormalization. In 1D, renormalization can be understood as removing some degrees of freedom by reducing the number of vertices. The basic renormalization process is known as (2-1) Pachner move of triangulation of 1D manifold.
To be more precise, the (2-1) move is an SLU transformation between two different 1D graphs: We note that the |G| is the order of the group G and we introduce the normalization factor 1/|G| 1/2 in the above expression due to the change of vertex number. Here, ν 2 (g 0 , g 1 , g 2 ) is a U (1)-valued function with variables g i ∈ G. Since we are constructing symmetric state, ν 2 should be symmetric under the action of G with ν 2 (gg 0 , gg 1 , gg 2 ) = ν 2 (g 0 , g 1 , g 2 ) (We note that ν 2 (gg 0 , gg 1 , gg 2 ) = ν * 2 (g 0 , g 1 , g 2 ) if g is anti-unitary). Since we are constructing fixed-point wave function, it should be invariant under renormalization. For instance, we can use two different sequences of the above (2-1) moves Eq. (22) to connect a fixed initial state and a fixed final state. Different approaches should give rise to the same wave function. These constraints give us the consistent equations for ν 2 .
The simplest example is the following two paths between two fixed states: The constraint is that the products of F moves for the above two processes equal to each other: The above equation implies: The above equation is exactly the same as the cocycle equation of group cohomology theory, and it means ν 2 should be a U (1) T -valued 2-cocycle.
Using an SLU transformation, we can further redefine the basis state |{g l } as In the new basis, one find that the phase factor in Eq. (22) becomes Since our gapped phases are defined by SLU transformations, ν 2 and ν 2 belong to the same phase. In general, the elements ν 2 in the same group cohomology class in H 2 (G, U (1) T ) correspond to the same 1D BSPT phase. SLU transformations not only give rise to the local rules of constructing fixed-point wave functions, but also give rise to commuting projector parent Hamiltonian for these fixed-point wave functions.
For example, in 1D, the parent Hamiltonian can be expressed as H = − i H i where the matrix element of H i are defined as: which only acts the states on site i and its neighboring sites. However, H i will not alter the states on neighboring sites of i. The above amplitude can be computed by SLU transformations by considering the following moves for a three site patch: C. Fixed-point wave function and classification for BSPT phases in 2D The fixed-point wave functions for BSPT phases in 2D are similar to the 1D case. We can again use the group element basis to construct the local Hilbert space on each vertex of arbitrary triangulation.
We assume that the triangulation admits a branching structure that can be labeled by a set of local arrows on all links (edges) with no oriented loop for any triangle. Mathematically, the branching structure can be regarded as a discrete version of a spin c structure and can be consistently defined on arbitrary triangulation of orientable manifolds. The basic renormalization process is known as (2-2) and (2-0)/(0-2) Pachner move of triangulation of 2D manifold. Moreover, according to the definition of BSPT phases, we also require that the support space of SLU transformations to be one-dimensional, such that it can adiabatically connect to a product state in the absence of global symmetry. Below, we will discuss physically consistent conditions for those SLU transformations generating fixed point wave functions.
We also add a normalization factor |G| −1/2 in the front of the (2-0) move operator, for the vertex number is reduced by one from the left state to the right state. [70] It is easy to check that the other (2-2) moves with different branching structure, e.g, the analogy of H-move, can always be derived by the standard (2-2)-move and (2-0)/(0-2) move. Considering the SLU transformation for the following patch: Thus, we conclude that The rest (2-2) moves are the analogies of dual F-move and dual-H move, they can also be derived from the basis (2-2) move and (2-0)/(0-2) move. For example, let us assume: whereν 3 (g 0 , g 1 , g 2 , g 3 ) is another U (1)-valued function which is different from ν 3 (g 0 , g 1 , g 2 , g 3 ). Considering the SLU transformation on the following patch: On the other hand, by applying the (2-0) move directly, we have Thus, we conclude thatν 3 (g 0 , g 1 , g 2 , g 3 )ν 3 (g 0 , g 1 , g 2 , g 3 ) = 1 orν 3 (g 0 , g 1 , g 2 , g 3 ) = ν −1 3 (g 0 , g 1 , g 2 , g 3 ). Moreover, the combination of (2-2) move and (2-0) move will further allow us to define a new set of renormalization move which reduces the number of vertices, namely, the (3-1) move. For example, considering the SLU transformation for the following patch, In Fig. 16 of the Appendix, we list all possible (2-2) and (3-1) moves that are consistent with a branching structure.
where s ijk = ±1 denotes the orientation of the triangle ijk . One finds that the phase factor in Eq. (34) becomes So the elements ν 3 in the same group cohomology class in H 3 (G, U (1) T ) correspond to the same 2D BSPT phase. Again, similar to the 1D case, the 2D SLU transformations can also be used to construct commuting projector parent Hamiltonian for these fixed point wave functions. The Hamiltonian term is a sequence of moves that change the group element of one vertex from g * to another g * . For example, we can consider the following moves for a triangular lattice: We have shifted the lattice a little, such that the branching structure is induced by a time direction from left to right. The first step of the above figures is a combination of three (2-2) moves. The second step is a (3-1) move that removes the vertex with group label g * at the center. The third step is a (1-3) move that create a vertex with group label g * at the center. And the last step is a combination of three (2-2) moves that change the lattice to the original shape. Since our wave function is at the fixed-point, the terms for different vertices commute with each other.

D. Fixed-point wave function and classification for BSPT phases in 3D
The fixed-point wave functions for BSPT phases in 3D are similar to the 1D and 2D cases. We can again use the group element basis to construct the local Hilbert space on each vertex of arbitrary triangulation. The basic renormalization process is known as (2-3) and (2-0) Pachner move of triangulation of 3D manifold.
Similar to the 1D and 2D cases, we can use SLU to redefine the basis state |{g l } as one finds that the phase factor in Eq. (34) becomes ν 4 (g 0 , g 1 , g 2 , g 3 , g 4 ) = ν 4 (g 0 , g 1 , g 2 , g 3 , g 4 ) µ 3 (g 1 , g 2 , g 3 , g 4 )µ 3 (g 0 , g 1 , g 3 , g 4 )µ 3 (g 0 , g 1 , g 2 , g 3 ) µ 3 (g 0 , g 2 , g 3 , g 4 )µ 3 (g 0 , g 1 , g 2 , g 4 ) , So the elements ν 4 in the same group cohomology class in H 4 (G, U (1) T ) correspond to the same 3D BSPT phase. We can also use the above moves to construct a 3D commuting projector parent Hamiltonian. Each term of the Hamiltonian is a sequence of 3D moves that changes the label of a vertex from g * to g * . All the terms commute with each other because the wave function is at the fixed-point.
Finally, we notice that for anti-unitary symmetry, e.g, time reversal symmetry, the above construction and classification scheme is not complete. It has been pointed out [8] that the decoration of E 8 state on the G-symmetry domain walls will give rise to new BSPT states beyond group cohomology classification. Apparently, the data H 1 (G, Z T ) classifies such a decorating pattern and the corresponding additional BSPT states. Since H 1 (G, Z T ) is trivial for all unitary symmetry group G and H 1 (Z T 2 , Z T ) = Z 2 for the (anti-unitary) time-reversal symmetry, we understand why the beyond group cohomology BSPT phases only arise for anti-unitary symmetry. Thus, we conclude that the two cohomology groups of the symmetry group G: H 1 (G, Z T ) and H 4 (G, U (1) T ) give rise to a complete classification of BSPT phases in 3D.

A. Fermionic symmetric local unitary transformations
In Ref. 72, it was shown that fermionic local unitary (FLU) transformations can be used to define and classify intrinsic topological phases for interacting fermion systems. The Fock space structure and fermion parity conservation symmetry of fermion systems can be naturally encoded into FLU transformations. It is well-known that the finite-time FLU evolution is closely related to fermionic quantum circuits with finite depth, which is defined through piecewise FLU operators. A piecewise FLU operator has the form U pwl = i e −iH f (i) ≡ i U (i), where H f (i) is a fermionic Hermitian operator and U (i) is the corresponding fermionic unitary operator defined in Fock space that preserve fermion parity (e.g., contains even number of fermion creation and annihilation operators) and act on a region labeled by i. Note that regions labeled by different i's are not overlapping. We further require that the size of each region is less than some finite number l. The unitary operator U pwl defined in this way is called a piecewise fermionic local unitary operator with range l. A fermion quantum circuit with depth M is given by the product of M piecewise fermionic local unitary operators: pwl . It is believed that any FLU evolution can be simulated with a constant depth fermionic quantum circuit and vice versa. Therefore, the equivalence relation between gapped states in interacting fermion systems can be rewritten in terms of constant depth fermionic quantum circuits: Thus, we can use the term FLU transformation to refer to both FLU evolution and constant depth fermionic quantum circuit. From the definition of FSPT state, it is easy to see that (in the absence of global symmetry): Namely, an FSPT state can be connected to a trivial state (e.g., a product state) vial FLU transformation (in the absence of global symmetry). Similar to the BSPT case, Eq. (54) implies that the support space of any FSPT in region A must be one dimensional. This is simply because a trivial state (e.g., a product state) has a one dimensional support space, and any FSPT state will become a product state via a proper local basis change (induced by a FLU transformation).
In the presence of global symmetry, we can further introduce the notion of invertable fermionic symmetric local unitary (FSLU) transformations to define and classify FSPT phases in interacting fermion systems. By FSLU transformation, we mean that the corresponding piecewise FLU operator is invariant under total symmetry group G f .

B. Layers of degrees of freedom
The are at most four layers of degrees of freedom in total in our fixed-point wave function of FSPT state (up to four spacetime dimensions). The bosonic states are always at the vertices. And the fermionic degrees of freedom (complex fermions, Majorana fermions and 2D p + ip chiral superconductors) are decorated on the intersecting sub-manifold of the bosonic state. In summary, the degrees of freedom of our FSPT states are [73] • |G b | level bosonic (spin) state |g i (g i ∈ G b ) on each vertex i.
• |G b | species of 2D p+ip chiral superconductors (may have several copies) on the dual surface of each codimension-2 simplex. The chiral Majorana modes along the edge of the dual surface are labelled by ψ σ ij...k,L or ψ σ ij...k,R depending on the chirality.
The above degrees of freedom has different symmetry transformation rules. The symmetry transformation of G b on bosonic state is the same as that in the BSPT states (g ∈ G b ): For complex fermions, we choose the symmetry transformations under G b to be The symmetry transformation rules of Majorana fermions γ σ ij...k,A/B are induced by the transformation of complex fermion a σ ij...k : And the symmetry transformations of chiral Majorana modes on the boundary of decorated p + ip superconductors are chosen to be We will discuss more about why we choose the transformation rules for Majorana modes in section VI F 2.
In this way, the G b species of fermions span a space that support a projective representation of G b with coefficient P ω2 f : with P f = −1 when acting on fermion parity odd states. We note that the projective representation U of G b is equivalent to a linear representationŨ of G f byŨ One can check directly thatŨ is indeed a genuine linear representation of G f : where the dot product in G f is defined in Eq. (2). In the previous constructions of FSPT state for G f = Z f 2 × G b , we put only one species of fermions on each simplex [52,53]. They transform trivially under the action of G b (and G f ). To construct FSPT state for G f = Z f 2 × ω2 G b , we need projective representation of G b with coefficient ω 2 to make it a linear representation of G f . We have chosen the canonical |G b |-dimensional projective representation Eq. (56), which can be constructed for arbitrary finite symmetry group G b . Although there are |G b | species fermions c σ ij...k (σ ∈ G b ) on each simplex ij...k , (at most) only one of them c gi ij...k will be decorated or in the occupied state, and all others fermion species c σ ij...k (σ = g i ) are in the vacuum states. In such a way, the FSPT constructions for symmetry group G f = Z f 2 × G b can be generalized to the case of

C. Symmetry conditions and consistency equations
Since we are constructing FSPT states, the F moves should be compatible with the symmetry action U (g) defined in Section III B. To be more precise, let us consider the following two dimensional commuting digram: where the horizontal F move changes the triangulations of the spacial manifold, and the vertical symmetry action U (g) changes the bosonic degrees of freedom from {g i } to {gg i } (similar for the fermionic ones). The outside parts "..." correspond to other triangulations of the spacial manifold and other symmetry actions on the states. This two dimensional diagram should commute for arbitrary horizontal Pachner move of arbitrary triangulations and vertical symmetry action with arbitrary g ∈ G b . The requirement of symmetric fixed-point wave function implies the following conditions: 1. The diagram Eq. (65) commutes, i.e., For the triangulations of d-dimensional space manifold, the Pachner move involves d + 2 vertices. So the basic F move can be denoted by F (g 0 , g 1 , ..., g d+1 ). With the help of Eq. (66), we can obtain the generic F move F (g 0 , g 1 , ..., g d+1 ) = g0 F (e, g −1 0 g 1 , ..., g −1 provided that we have defined the standard F symbol with the first argument being the identity element e ∈ G b . Using this definition of F move, Eq. (66) is automatically satisfied. This is because of the following commuting diagram We can deduce the dashed arrow U (g) from solid arrows U (g 0 ) and U (gg 0 ), due to U (g)U (h) = P ω2(g,h) f U (gh) and the fermion parity even property of the F operators.

IV. FIXED-POINT WAVE FUNCTION AND CLASSIFICATION OF FSPT STATES IN 1D
In this section, we give the explicit constructions and classifications of 1D FSPT states. The fixed-point wave functions are obtained by decorating complex fermions to BSPT states consistently. Formally, the wave function is a superposition of all basis states |{g i } with legitimate decorations: The basis state |{g i } is a state (with vertex i labelled by g i ∈ G b ) decorated by complex fermions c σ ij at link ij . The constructed fixed-point wave function |Ψ should be both symmetric and topological (invariant under retriangulations of the lattice). As will shown below, these constraints would give us the consistency conditions for the 1D FSPT classifications summarized in Eqs. (8) - (11).
We note that the 1D Kitaev chain is a fermionic invertible topological order. Since it does not need any bosonic symmetry protection (Z f 2 can not be broken), we do not consider it as FSPT state. The Kitaev chain layer is useful when considering it as the ASPT on the boundary of a 2D FSPT state. The 2D classification data n 2 will be trivialized.
This section will be organized as follows. The two layers of degrees of freedom (bosonic states and complex fermions) are introduced in section IV A. In section IV B, we propose the procedures of symmetrically decorating complex and Majorana fermions to BSPT states. Then the construction and the consistency equations of the FSLU transformations are discussed in sections IV C and IV D, respectively.

A. Two layers of degrees of freedom
The basic idea to construct FSPT states is to decorate complex fermions to the BSPT states. Therefore, there are two layers of degrees of freedom, including the bosonic ones, in the 1D lattice model: at the center of each link ij . These degrees of freedom are summarized in one unit cell in the following figure: Here, we choose the link direction from vertex i to vertex j (from left to right). The vertices are labelled by g i and g j which are elements of G b . Blue ball is the decorated complex fermion c σ ij (σ ∈ G b ) at the center of link ij . The symmetry transformations of these degrees of freedom under G b are the same as the discussions in section III B. To be more specific to the 1D case, we summarize them as (g, g i , σ ∈ G b ): While the bosonic degrees of freedom on each vertex form a linear representation of G b , the complex fermions form projective representation of G b with coefficient (−1) ω2 . In this way, they all transform linearly under the action of G f defined by Eq. (63). Although there are |G b | species of complex fermions in the Hilbert space of the system, we will see later that (at most) only one of them is decorated nontrivially in the fixed-point wave function. If we consider the case of ω 2 = 0 (i.e., G f = Z f 2 × G b ), the symmetry transformation rules are independent of group element label σ of the fermions [see Eq. (72)]. Therefore, we can reduce these |G b | species of fermions to only one species without group element label. The resulting states are exactly the ones studied in Refs. 52 and 53.

B. Decoration of complex fermions
In the group cohomology theory of BSPT phases [4], the fixed-point wave functions are constructed as superpositions of all basis state |{g i } . The coefficients in front of these basis states are U (1)-valued cocycles. To construct FSPT states, we have introduced the fermionic degrees of freedom associated to the basis states in the previous section. In the following, we would discuss the detailed procedures of systematically decorating complex fermions. These decorations should be designed to be symmetric under symmetry actions.
The complex fermion decoration is specified by a Z 2 -valued 1-cochain n 1 ∈ C 1 (G b , Z 2 ), which is the first classification data for 1D FSPT phases. If n 1 (g i , g j ) = 0, all the modes of complex fermions c σ ij (σ ∈ G b ) at link ij are unoccupied (shown by blue circles in figures). On the other hand, if n 1 (g i , g j ) = 1, exactly one complex fermion c gi ij will be decorated at the center of the oriented link ij (shown by filled blue balls in figures). And all other complex fermions c σ ij (σ = g i ) are still in vacuum states. The above complex fermion decoration rule is G b -symmetric. Under a U (g)-action, the vertex labels of link ij become gg i and gg j . According to the decoration rule, the decorated complex fermion [if n 1 (gg i , gg j ) = n 1 (g i , g j ) = 1] should be c ggi ij , which is exactly the complex fermion c gi ij by a U (g)-action.

C. F moves
For a fixed triangulation of spacial manifold, we can decorate complex fermions symmetrically as discussed above. However, we want to construct fixed-point wave functions that are invariant under retriangulation of the space. To connect different triangulations, there are FSLU transformations for each Pachner move. For the 1D lattice, there is essentially only one Pachner move given by where the FSLU F operator is defined as In the above expression of F symbol, |G b | 1/2 is the normalization factor because the number of the lattice sites is reduced by one. ν 2 (g 0 , g 1 , g 2 ) is a U (1) phase factor depending on three group elements of G b . For BSPT states, the F operator has only these two bosonic factors. For FSPT states, however, there are complex fermion term of the form c † cc. The complex fermion term annihilate the possibly decorated (depending on n 1 ) complex fermions c g1 12 and c g0 01 on the two links in the right-hand-side state, and create a new complex fermion c g0 02 at the center of link 02 in the left-hand-side state.
As discussed above, the F move should be a FSLU operator. Therefore, it should be both fermion parity even and symmetric under G b -action. These two conditions give us several consistency equations for the classification data n 1 and ν 2 .

Fermion parity conservation
Since the complex fermions are decorated according to n 1 (g i , g j ), the complex fermion parity change of the F move Eq. (73) is As a result, the conservation of fermion parity under the F move enforces the condition which is the cocycle equation for the decoration data n 1 .

Symmetry condition
The F move should also be consistent with the symmetry actions [see Eq. (65)]. In 1D, we have the following commuting diagram or the symmetry condition for F operators As discussed in Section III C, the above equation can be viewed as a definition of the generic F (g 0 , g 1 , g 2 ) in terms of the standard F move F (e, g −1 0 g 1 , g −1 0 g 2 ): Therefore, we only need to fix the expression of the standard F move F (e, g −1 0 g 1 , g −1 0 g 2 ), and all other non-standard F moves are obtained by a symmetry action on the standard one. The explicit expression of the standard F move F (e, g −1 0 g 1 , g −1 0 g 2 ) is given by Note that the U (1) coefficient in the standard F move is chosen to be the inhomogeneous cochain ν 2 (g −1 And we do not impose the condition "ν 2 (gg 0 , gg 1 , gg 2 ) = ν 2 (g 0 , g 1 , g 2 )" in priori. In fact, as shown below, this condition does not hold in general.
We can apply a U (g 0 )-action on the standard F move Eq. (80), and compare it with the generic expression Eq. (74). The symmetry conditions for n 1 and ν 2 are where the term (−1) ω2 n1 comes from the symmetry transformations of c We also introduced new notations to relate the homogeneous cochain ν 2 (g 0 , g 1 , g 2 ) = g0 ν 2 (g −1 0 g 1 , g −1 1 g 2 ) and the inhomogeneous cochain In the following, when we write the cochain ν d without arguments, we will always mean the inhomogeneous one, i.e., ν d = e ν d .

D. Associativity and twisted cocycle equations
The F move reduces three vertices on the lattice to two vertices. If one consider reducing four vertices to two vertices, there are two inequivalent ways to do that. The final results should be independent of the two ways. This gives us the consistency equation for Pachner moves: In terms of F operators, the above commuting diagram means Similar to the standard F symbol which can be used to derive all other non-standard ones by a symmetry action, we can also assume g 0 = e in the above equation. All other consistency equations with generic g 0 can be deduced from this standard equation by a U (g 0 )-action. Therefore, we only need to consider the consistency equation The above equation is simpler than the generic one Eq. (84), since only the last F symbol is non-standard. Substituting the standard F move Eq. (80), the consistency equation Eq. (85) becomes where the last term (−1) ω2 n1 comes from the U (g −1 0 g 1 )-action on F (e, g −1 (82)]. Note that the complex fermions do not contribute any fermion signs. So we have the twisted cocycle equation for inhomogeneous ν 2 : with obstruction function In summary, the associativity condition for the F moves in 1D gives us the twisted cocycle equation Eqs. (87) and (88) for inhomogeneous cochain ν 2 .

E. Classification of 1D FSPT phases
The general classification of 1D FSPT phases is as follow. We first calculate the cohomology groups H 1 (G b , Z 2 ) and H 2 (G b , U (1) T ). For each n 1 ∈ H 1 (G b , Z 2 ), we solve the twisted cocycle equation Eq. (10) for ν 2 . If ν 2 is in the trivialization subgroup Γ 2 in Eq. (11), it is known to be trivialized by complex fermion decoration [52], see Appendix A for more details. So the obstruction-free n 1 and trivialization-free ν 2 fully classify the 1D FSPT phases.
We note that we can use the FSLU transformations to construct the commuting projector parent Hamiltonians. Each term of the Hamiltonian is a sequence of fermionic F moves that changes the label of a vertex from g * to g * . All the terms commute with each other for our FSPT wave function is at the fixed-point.

V. FIXED-POINT WAVE FUNCTION AND CLASSIFICATION OF FSPT STATES IN 2D
In this section, we construct and classify FSPT states in two spacial dimensions. The fixed-point wave function is again a superposition of all basis states |{g i } with fermion decorations: The basis state |{g i } is a state (with vertex i labelled by g i ∈ G b ) decorated by complex fermions c σ ij at link ij and Majorana fermions γ σ i,A and γ σ i,B near vertex i according to several designed rules. So the fixed-point wave function would looks like A. Three layers of degrees of freedom In 2D, we decorate two layers of fermionic degrees of freedom to the BSPT states. Therefore, there are three layers of degrees of freedom, including the bosonic ones, in our 2D triangulation lattice model: These three layers of degrees of freedom are summarized in one triangle in the following figure: Here, the three vertices of the triangle are labelled by g 0 , g 1 , g 2 ∈ G b . Blue ball is the complex fermion c σ 012 (σ ∈ G b ) at the center of triangle 012 . Red dots represent Majorana fermions γ σ ij,A and γ σ ij,B (σ ∈ G b ) on the two sides of link ij .
The symmetry transformations of these degrees of freedom under G b are summarize them as follows (g, g i , σ ∈ G b ): As in the 1D case, the bosonic degrees of freedom form a linear representation of G b (and G f ). On the other hand, the fermion modes support projective representations of G b with coefficient (−1) ω2 , and linear representations of G f defined by Eq. (63).
In the simpler case of G f = Z f 2 × G b , all the flavors of fermions have the same transformation rule for different group element label σ. So we can suppress the species labels. This is again the previous group supercohomology models [52,53].

B. Decorations of fermion layers
In this section, we would give a systematic procedure to decorate Kitaev chains and complex fermions to the basis state |{g i } labelled by g i ∈ G b for each vertex i. Similar to the 1D case, we only decorate (at most) one species of fermions to the state, although the Hilbert space is spanned by |G b | copies of fermions. Again, the decorations should be designed to respect the symmetry.

Kitaev chain decoration
The Kitaev chain decoration in 2D is similar to the constructions in the pioneering works Refs. 74 and 75. However, we will adopt the more general procedures in Ref. 53, which can deal with arbitrary triangulations of the 2D spacial manifold. The generalization in this paper for symmetry group G f = Z f 2 × ω2 G b is that we put (at most) one of the G b species Majorana fermions into nontrivial pairings and all others vacuum pairings. If we consider the symmetry group G b = Z T 2 and nontrivial 2-cocycle ω 2 (T, T ) = 1, our construction on a fixed triangular lattice will reproduce to the exactly solvable T 2 = −1 topological superconductor model in Ref. 76.
To simplify our notations and make it easier to generalize to higher dimensions, we present some notations for Majorana fermion pairings. For two Majorana fermions γ e i,C and γ g −1 h j,D at vertices i and j (g, h ∈ G b and C, D = A, B), we can choose the pairing such that when acting on this state. We will call it standard pairing, as the first Majorana fermion is labelled by the identity element e ∈ G b . The standard pairing will be illustrated in figures by a red arrow pointing from vertex iC to vertex jD. For the non-standard pairing between γ g i,C and γ h j,D , we can use a U (g)-action on both sides of Eq. (96) and obtain where δ CB = 1 (= 0) if the Majorana fermion γ σ i,C is the B type (A type) one. This difference comes from the symmetry transformations of A and B type Majorana fermions [see Eqs. (94) and (95)]. For simplicity in describing the pairing, we introduce the projection operator of the Majorana fermion pairing as This generic pairing projection operator P g,h iC,jD is obtained from a U (g)-action on the standard pairing projection So the symmetric nature of the pairings can be easily seen from the symmetry transformations of the projection operators (g, h, k ∈ G b ): In 2D and higher dimensions, we will use the generic pairing rule Eq. (97) and the projection operator Eq. (98) to construct G b -symmetric states. a. Decoration procedure. Our Majorana fermions γ σ ij,A and γ σ ij,B (σ ∈ G b ) are on the two sides of each link ij . We use the convention that the Majorana fermion on left-hand-side (right-hand-side) of the oriented link ij is γ σ ij,A (γ σ ij,B ). The vacuum pairing between them is from A to B: −iγ σ ij,A γ σ ij,B = 1. To decorate Kitaev chains on the lattice, we should also add arrows to the small red triangle inside each triangle 012 (see Fig. 2). These red arrows are constructed from the discrete spin structures (a choose of trivialization of Stiefel-Whitney homology class w 0 dual to cohomology class w 2 ) of the 2D spacial spin manifold triangulation. The Majorana fermions are designed to pair up with each other according to these red arrows. The red arrows constructed have the property that the number of counterclockwise arrows in a loop with even red links is always odd. This is crucial for the decorated Kitaev chain to have fixed fermion parity. For details of the Kasteleyn orientations for arbitrary triangulation, we refer the interested readers to Ref. 53. The Kitaev chain decoration is specified by n 1 (g i , g j ) ∈ Z 2 , which is a function of two group elements g i , g j ∈ G b . If n 1 (g i , g j ) = 0, the Majorana fermions γ σ ij,A and γ σ ij,B on the two sides of link ij are in vacuum pairings: Here is an example of the Kitaev chain decoration around the vertex g 2 inside a triangle (we omit the operator labels of Majorana fermions which are in vacuum pairings): The domain wall decorated by a Kitaev chain is indicated by a green belt. Trivial (vacuum) pairings and nontrivial pairings are represented by dashed red lines and solid red lines, respectively. And the red (blue) arrows show that the trivial (nontrivial) pairing directions of Majorana fermions: We will discuss more about the pairing directions and why they are symmetric later. b. Consistency condition. According to our decoration rule, the number of decorated Kitaev chains going though the boundary of a given triangle 012 is (dn 1 )(g 0 , g 1 , g 2 ) = n 1 (g 1 , g 2 ) + n 1 (g 0 , g 2 ) + n 1 (g 0 , g 1 ). (103) Since we are constructing gapped state without intrinsic topological order, there should be no dangling free Majorana fermions inside any triangle. Therefore, we have the (mod 2) equation This is the consistency condition for the Kitaev chain decoration data n 1 .
c. Symmetric pairing directions. Now let us turn back to the details of Majorana fermion pairings inside the triangle 012 . The strategy of constructing G b -symmetric pairings is the same as in the 1D case: we first consider the standard triangle of g 0 = e, and then apply a U (g 0 )-action to obtain all other non-standard triangles. The Majorana fermion pairings constructed in this way are automatically symmetric, due to the symmetry transformation rule of the pairing projection operators Eq. (99). For the standard triangle, the Majorana fermions are paired (trivially or nontrivially) according to the Kasteleyn orientations indicated by red arrows. The pairings in the non-standard triangle is obtained by a U (g 0 )-action as follows: Note that Majorana fermions γ σ ij,A and γ σ ij,B (σ = g i ) of link ij are always in vacuum pairings −iγ σ ij,A γ σ ij,B = 1 , independent of the n 1 configurations. So their pairing directions always follow the red arrow Kasteleyn orientations in both figures of the above equation. For the two Majorana fermions γ gi ij,A and γ gi ij,B of the link ij , there are two possibilities. If n 1 (g i , g j ) = 0, these two Majorana fermions are also in vacuum pairing, with direction indicated by the red arrow and projection operator On the other hand, if n 1 (g i , g j ) = 1, we will pair the Majorana fermion inside the triangle with another one belonging to another link with also n 1 = 1 [for example, γ g0 02B and γ g1 12A are paired in Eq. (100)]. Note that there are always even number of Majorana fermions in nontrivial pairing among the three ones (γ g1 12A , γ g0 02B and γ g0 01A ) inside the triangle 012 , for we have (dn 1 )(g 0 , g 1 , g 2 ) = 0 (mod 2) from Eq. (104). There are three possible nontrivial pairings inside the triangle 012 , with Majorana pairing projection operators Among the three possible nontrivial pairings, only the last two may change their directions in the non-standard triangle. They are indicated by blue arrows in the right-hand-side figure of Eq. (105). This can be understood from the following facts: The (−1) ω2 term appears in the projection operators when the pairing is between Majorana fermions with different group element labels; And the (−1) s1 term appears when the pairing is between the same A/B type Majorana fermions. The pairing Eq. (107) between γ g0 02B and γ g0 01A belongs to neither of the above two cases. So their pairing direction is the same as the red arrow even after U (g 0 )-action.
d. Majorana fermion parity. Since the symmetry action may change the pairing directions inside a triangle, the Majorana fermion parity of this triangle may also be changed. The fermion parity difference between the standard and non-standard triangles can be calculated from the number of pairing arrows that are reversed by U (g 0 )-action, which of course depends on the n 1 configurations. We can use, for example, n 1 (g 0 , g 1 )n 1 (g 1 , g 2 ) = 0, 1 to indicate whether γ g1 12A and γ g0 01A are paired or not. So the Majorana fermion parity change inside the triangle is in general given by where we have used (dn 1 )(g 0 , g 1 , g 2 ) = 0 from Eq. (104) in the second step. The above equation is a summary of phase factors from Eqs. (107)-(109). We note that the above expression is also true for negative oriented triangles. All the fermion parity change cases involve the particular Majorana fermion γ g1 12A (γ g1 12B for negative oriented triangles). We will use it later in the definition of F symbol to compensate the fermion parity changes of the Majorana fermion pairing projection operators.
Although the Majorana fermion parity of a given triangle may be changed, the fermion parity of the whole system is fixed under the global U (g)-action. Since the fermion parity of the vacuum pairings are not changed by symmetry actions, we only need to consider the n 1 domain walls decorated by Kitaev chains. For a particular (closed) Kitaev chain, the Majorana fermion parity is the same as the vacuum pairings if the pairings are constructed according to Kasteleyn orientations of the resolved dual lattice. It is also not hard to show that symmetry action will always change the arrow even times, following the pairing rules of Eq. (105). Therefore, we conclude all closed Kitaev chains will have even fermion parity, although the local fermion parity of a triangle may be changed compare to the Kasteleyn orientations.
To sum up, among the |G b | species of Majorana fermions, we decorate exactly one Kitaev chain to each symmetry domain wall specified by the n 1 configurations of the state. The decoration is symmetric under symmetry actions. The Majorana fermion parity of a triangle is changed according to Eq. (110) compared to the Kasteleyn oriented pairings.

Complex fermion decoration
The rules of complex fermion decoration are much simpler than the pairings of Majorana fermions. The decoration is specified by a Z 2 -valued 2-cochain n 2 ∈ C 1 (G b , Z 2 ). If n 2 (g i , g j , g k ) = 0, all the modes of complex fermions c σ ijk (σ ∈ G b ) at the center of triangle ijk (i < j < k) are unoccupied. On the other hand, if n 2 (g i , g j , g k ) = 1, exactly one complex fermion mode c gi ijk will be decorated at the center of triangle ijk . All other complex fermions c σ ijk , which is exactly the complex fermion c gi ijk by a U (g)-action.

C. F moves
To compare the states on different triangulations of the 2D spacial manifold, we should consider the 2D Pachner move, which is essentially the retriangulation of a rectangle. The Pachner move induces a FSLU transformation of the FSPT wave functions from the right-hand-side triangulation lattice T to the left-hand-side lattice T . We can first define the standard F move for rectangle with g 0 = e, then other non-standard ones can be obtained by simply a U (g 0 )-action. The standard F move is given by: where the FSLU F operator is defined as We used the abbreviationīj for g −1 i g j in the arguments of F . And n 2 (ijk) represents n 2 (g i , g j , g k ) = n 2 (g −1 i g j , g −1 j g k ) for short.
The U (1) phase factor ν 3 (01,12,23) = ν 3 (g −1 0 g 1 , g −1 1 g 2 , g −1 2 g 3 ) in the front of F symbol is a inhomogeneous 3-cochain depending on three group elements. By definition, it is related to the homogeneous cochain by with the first argument of homogeneous cochain to be the identity element e ∈ G b . Later, we will use symmetry conditions to relate ν 3 (e, g −1 0 g 1 , g −1 0 g 2 , g −1 0 g 3 ) and ν 3 (g 0 , g 1 , g 2 , g 3 ). The complex fermion term of the form c † c † cc annihilate two complex fermions at the two triangles of the righthand-side figure and create two on the left-hand-side figure of Eq. (111). According to our decoration rules developed in section V B 2, the triangle ijk is decorated by complex fermion c gi ijk . So in the standard F move, only the last fermion c g −1 0 g1 123 has group element label g −1 0 g 1 , and the other three fermions all have group element label e. We note that, different from the [53], the complex fermion parity of the 2D F move does not need to be conserved in general.
The term X 0123 [n 1 ] is related to the Kitaev chain decorations. In terms of Majorana fermion pairing projection operators Eq. (98), the general expression of X operator is The Majorana fermion γ is added for fermion parity considerations (which will be discussed in detail in the next section). If we do not add this term, X 0123 [n 1 ] would project the right-hand-side state to zero, whenever the Majorana fermion parity is changed under this F move. We choose the Majorana fermion γ  111) is Since there is no pairings for the two blue arrow links in Eq. (111), the Majorana fermion parity is always conserved for this n 1 configuration.
The F symbol constructed above should be a FSLU operator. So it should be both fermion parity even and symmetric under G b -action. Similar to the 1D case, we can use these conditions to obtain several consistency equations for the cochains n 1 , n 2 and ν 3 .

Fermion parity conservation
It is proved that the Majorana fermion parity is conserved under 2D F move if they are paired according to the Kasteleyn orientations in 2D [53]. Nevertheless, some of the links are not Kasteleyn oriented in the standard F move Eq. (111). This is because the triangle 123 is non-standard, i.e., the group element label of the first vertex is not e ∈ G b . It should be obtained from the standard one by a U (g −1 0 g 1 )-action. So the blue arrows inside this triangle may change their directions according to our symmetric pairing rules. The Majorana fermion parity change of this triangle can be calculated from Eq. (110). Note that the three group element labels of the vertices are now g −1 0 g 1 , g −1 0 g 2 and g −1 0 g 3 . So the Majorana fermion parity change under the standard F move is On the other hand, the complex fermion parity change under the F move can be simply calculated by counting the complex fermion numbers of the two sides: As a result, the conservation of total fermion parity ∆P f = ∆P γ f · ∆P c f = 1 under the F move enforces the condition It shows that the Majorana fermions and complex fermions are coupled to each other. This is very different from the 2D FSPT states with unitary group G f = Z f 2 × G b (i.e., ω 2 = 0 and s 1 = 0) [53], where the fermion parities of the Majorana fermions and complex fermions are conserved separately. So Eq. (119) is reduced to a simple cocycle equation dn 2 = 0. In the case of T 2 = −1 topological superconductors [76], although both ω 2 and s 1 are nontrivial, there combination ω 2 + s 1 n 1 = 0 is also trivial. So we still have dn 2 = 0. That is the reason why it admits exactly solvable model with only Kitaev chain decorations.

Symmetry condition
In the previous discussions, we only considered the standard F move with g 0 = e. The non-standard F moves are constructed by symmetry actions on the standard one. From Eq. (65), we have the following commuting diagram: So the non-standard F operator is defined as The F moves constructed in this way are automatically symmetric, because we can derive the transformation rule using Eq. (62) and the fact F is fermion parity even. After a U (g 0 )-action on the standard F operator Eq. (112), we can obtain the generic F symbol expression: The complex fermions now have group element labels g 0 or g 1 . And the X operator is with added Majorana fermion γ g2 23B rather than γ 23B . P 0123 [n 1 ] has similar expression as Eq. (115) that project the Majorana fermions to the pairing state on the left-hand-side figure (the group element labels are changed appropriately).
From the decoration rules of Majorana fermions and complex fermions, n 1 and n 2 are invariant under symmetry actions. The generic homogeneous cochain ν 3 in Eq. (123) is a combination of the inhomogeneous ν 3 in the standard F move and the ±1 signs appearing in the symmetry action. So we have the following symmetry conditions for n 1 , n 2 and ν 3 : The symmetry sign O symm 4 in the last equation is given by where the sign (−1) ω2(g0,g −1 0 g1)n2(123) comes from the symmetry transformation Eq. (93) of c g −1 0 g1 123 , and the sign in the X operator. We note that the last term ω 2 dn 2 in Eq. (128) is not a cup product or cup-1 product form. This symmetry sign O symm 4 will appear later in the twisted cocycle equation for ν 3 as part of the obstruction function [see Eq. (131)].

D. Super pentagon and twisted cocycle equations
The F moves should satisfy a consistency condition known as pentagon equation for fusion categories. In fermionic setting, it is a super pentagon equation with some fermion sign twist for superfusion categories [72,77,78]. The 2D FSPT states correspond to a special kind of superfusion category in which all the simple objects are invertible. So the classification of 2D FSPT states can be understood mathematically as the classification of pointed superfusion categories corresponding to a given symmetry group.
Similar to previous discussions, we only need to consider the standard super pentagon equation with g 0 = e. All other super pentagon equations can be obtained from it by a U (g 0 ) symmetry action. So it is enough to merely consider the standard super pentagon as coherence conditions. This standard super pentagon equation is shown in Fig. 3. Algebraically, we have the following equation where we usedīj to denote g −1 i g j . Note that only the last F symbol is non-standard in the above equation. Now we can substitute the explicit expression of the standard F move Eq. (112) into the standard super pentagon equation Eq. (129). After eliminating all complex fermions and Majorana fermions, we would obtain a twisted cocycle equation for the inhomogeneous 3-cochain ν 3 . In general, the twisted cocycle equation reads where O 4 [n 2 ] is a functional of n 2 only (as well as ω 2 and s 1 parametrizing the symmetry group, of course These four terms have different physical meanings and are summarized as Note that the dn 2 terms in the exponent of (−i) in the last equation should be understood as taking mod 2 values (can only be 0 or 1). And the notation i is the higher cup product by Steenrod [65]. By adding a coboundary (−1) d(s1 n2+n2 2dn2 ) to the obstruction function and shifting ν 3 → ν 3 (−1) s1 n2+n2 2dn2 , we can simplify the above obstruction function to We note that only the first three terms ω 2 n 2 + n 2 n 2 + n 2 1 dn 2 in the exponent are expressed as (higher) cup product form, while other terms are not. If we consider the special case of ω 2 = 0 and s 1 = 0, then we have This is a generalization of the usual sign twist (−1) n2 n2 for 2-cocycle n 2 . If dn 2 = 0, we would have another term (−1) dn2 1n2 .
Since the fermion parities of the X operator and the complex fermion operator c † c † cc are only related to dn 2 , this obstruction function O cγ 4 [dn 2 ] is a functional of dn 2 (rather than n 2 directly). In the rest of this subsection, we would calculate the most complicated part O γ 4 [dn 2 ] of the obstruction function. In addition to ±1, this Majorana fermion term can also take values in ±i. Whenever the Majorana fermion parity of the F move is changed, i.e. dn 2 (0123) = 1, there will be a dangling Majorana fermion γ dn2(0124) and X 0123 = P 4 γ0 2 23B dn2(0123) .
Here, P i is the Majorana pairing projection operator of the corresponding i-th figure (1 ≤ i ≤ 5) in the super pentagon equation Fig. 3. We use the convention that the rightmost figure is the first one with projection operator P 1 , and the other four figures are counted counterclockwise. We also use the simpler notation Using these X operators, the obstruction function coming from Majorana fermions can be calculated by The average is taken over the Majorana fermion state of the rightmost figure in Fig. 3. We also inserted P 1 , which is 1 acting on the rightmost state, at the first and the last places of the operator string.
Eq. (140) should be calculated separately for different Majorana fermion configurations. Among the five dangling Majorana fermions of the five X operators, only three of them are different: whereî means that the number i is removed from the argument. Each element of the triple corresponds to the Majorana fermion parity change of one or several F moves. There are in total 2 3 /2 = 4 different possibilities for the Majorana fermion parity changes (see the first column of dn 2 (0)+dn 2 (2)+dn 2 (1) Note that we replaced P 1 by the second term of Eq. (144) to obtain Eq. (146) (see the third column of Table IV the exponent of (−1) in Eq. (147) The dn 2 terms in the exponent of (−i) should be understood as taking mod 2 values (can only be 0 or 1). This is exactly the result claimed previously in Eq. (135).

E. Boundary ASPT states in Γ 2
We have constructed 2D FSPT states in the above discussions. However, not all of them correspond to distinct FSPT phases. In the following subsection, we will construct explicitly a FSLU transformation path to connect an FSPT state with n 2 = ω 2 and a state without complex fermion decorations. The physical understanding is that there is a gapped symmetric boundary for the 2D FSPT state. So we conclude the state with n 2 , which is in the new coboundary subgroup should be considered as in the trivial FSPT phase.

FSLU to trivialize the 2D bulk
Let us fix the symmetry group G f with given G b , ω 2 and s 1 . We consider the special group supercohomology 2D FSPT state constructed from (n 2 , ν 3 ) data satisfying dn 2 = 0 and dν 3 = O 4 = (−1) n2 n2 [52]. We will show that the 2D FSPT state with n 2 = ω 2 can be connected to a product state by FSLU transformations.
Consider a 2D triangulation lattice of a closed oriented spacial manifold. The FSPT wave function is a superposition of the basis states with coefficients related to ν 3 [see Eq. (89)]. The n 2 data specifies the decorations of complex fermion c gi ijk at the center of each triangle ijk of the bosonic basis state |{g i } . So the fermionic basis state can be expressed as We will show in the following that the above state can be transformed by two FSLU as where the schematic figures of these three states are shown in Fig. 4. The final state |(c) is obtained from the bosonic state |{g i } by decorating a small Kitaev chain around each vertex (see the Kitaev chain along the gray arc in Fig. 4c).
After shrinking the small Kitaev chain to each vertex i, we can view the state |g i with a fermion mode as a new basis state. So the final state has the expression which is a fermionic product state without complex fermion decorations on the triangles. Therefore, using the two FSLU U 1 and U 2 , we have connect an FSPT state with complex fermion decorations specified by n 2 = ω 2 to another FSPT state without complex fermion decorations. So the decoration data n 2 = ω 2 for complex fermion layer is trivialized.  Fig. 4b are The blue arrow on the red link in Fig. 4b means that the actual arrow direction is obtained by a G b -action and depends on ω 2 . The above Kitaev chain decoration procedure along the boundary of a triangle is very similar to the 1D FSPT construction with only three vertices. The Majorana pairings have the following properties: (i) Both the vacuum and the nontrivial pairings are invariant under G b -action. (ii) The fermion parity change of Majorana chains from Fig. 4a to Fig. 4b inside the triangle is exactly (−1) ω2(g −1 0 g1,g −1 1 g2) . Since we have chosen n 2 = ω 2 , we conclude the complex fermion mode c gi ijk at the center of the triangle should be totally annihilated, to make sure that U 1 is fermion parity even.
In such a way, the FSLU transformation U 1 annihilate the fermions at the center of each triangle and create a small Kitaev's Majorana chain along the boundary of the triangle. The explicit expression of the standard FSLU transformation U 1 with g 0 = e is The first term c g k ijk is used to remove the complex fermions at the center of each triangle. And other terms are Majorana fermion pairing projection operators Eq. (98) to decorate one Kitaev chain along the boundary of the triangle. The last dangling Majorana operator is inserted for fermion parity considerations. Other non-standard U 1 can be obtained from the standard one by a U (g 0 )-action. Therefore, the operator U 1 from |(a) to |(b) is both fermion parity even and symmetric under G b -action.
(2) The second FSLU transformation U 2 from |(b) to |(c) . Since the state is on a closed oriented surface, there are four Majorana fermions on the two sides of the oriented link ij . For convenience, we now relabel them by γ gi ij1 , γ gj ij1 on the right-hand side, and γ gi ij2 , γ The expression for the FSLU U 2 is simply Note that the actual direction of the blue arrow in Fig. 4c near vertex i depends on P gi,gi ij1,ij2 = U (g i ) 1 2 (1 − iγ e ij1 γ e ij2 )U (g i ) † . So the arrow direction will be reversed if s 1 (g i ) = 1, and γ gi ij1 and γ gi ij2 are Majorana fermions belonging to the same A/B type.
After the above two FSLU transformations U 1 and U 2 , we have a state where each vertex is surrounded by one nontrivial Majorana chain (red arrows) and |G b | − 1 trivial Majorana chains (green arrows). The fermion parity of this vertex depends on the number and the orientations of the triangles sharing this vertex. We can define a new state |g i around the vertex i as the combination of the original bosonic state |g i and the neighboring Majorana fermions (the degrees of freedom inside the gray circles in Fig. 4c). It is easy to check that |g i has the same G b -transformation property as |g i , i.e., |g i → |gg i . So the final state |(c) is a fermionic product state Eq. (156).
In summary, using the two FSLU transformations Eqs. (160) and (161), we can remove the complex fermions at the triangles of an FSPT state with n 2 = ω 2 , and obtain an FSPT state with n 2 = 0. Therefore, the complex fermion decoration layer with n 2 = ω 2 is trivialized by these FSLU transformations.

Boundary ASPT of the 2D bulk
We have shown the 2D FSPT state with n 2 = ω 2 on a closed surface can be connected to an FSPT state with n 2 = 0. However, for a system with boundary, there is something unusual left.
Consider a state defined on a 2D triangulation lattice with boundary. We can perform FSLU transformations similar to Eqs. (160) and (161). The only difference is that the link ij in the product in Eq. (161) is not on the boundary of the space manifold, since there are only two Majorana fermions near the each boundary link (see Fig. 5 for an example with only one interior vertex labelled by g * ). After the transformations, the bulk state becomes a tensor product of interior vertex state |g i (see Fig. 6a for an example). But the boundary is only transformed under U 1 and is a so-called 1D ASPT state (see Fig. 6b for an example). This boundary state is again a combination of one nontrivial Majorana chain (see red links in Fig. 6b) and |G b | − 1 trivial ones (see green links in Fig. 6b).   The anomalous feature of the boundary can be seen from the symmetry action on the boundary. Under a U (g) symmetry action, the Majorana fermions are transformed as γ gi A → (−1) ω2(g,gi) γ ggi A , γ gi B → (−1) ω2(g,gi)+s1(g) γ ggi B . Since there are three types of Majorana pairings on the boundary (see Fig. 6b), we should analyze their symmetry transformations separately: Therefore, depending on ω 2 (g, g i ), ω 2 (g, g j ) and s 1 (g), the local Majorana fermion parity (pairing direction) for the second and third types pairings may be changed. For a closed 1D array of Majorana fermions as the boundary of a 2D bulk (see Fig. 6b for example), it is not hard to show that the total Majorana fermion parity is always fixed under G b -action [79]. However, if we want to define a similar state on an open chain (such as Fig. 6b with open boundary condition), the total fermion parity may be violated under U (g)-action. This is simply because the direction of the link crossing the boundary may be changed. So the symmetry action is incompatible with the fermion parity of the open 1D ASPT chain. It implies that the 1D ASPT state can exist only on the boundary of a 2D bulk state.

Boundary F move and fermion parity violation
There is another way to understand the anomalous feature of the 1D boundary. We can try to construct such 1D state without 2D bulk directly, and find out the inconsistency of the state.
We consider the 1D state with a Kitaev chain, and without complex fermion decoration (n 1 = 0). We put |G b | species of Majorana fermions near each vertex similar to the 2D construction. But only one of them is in nontrivial pairings between different vertices. The F move for this state is given by where the FSLU F operator is defined as X is some Majorana fermion projection operators to impose the Majorana pairings. Since the Majorana fermion parity change of the F move is (−1) ω2(g −1 0 g1,g −1 1 g2) (one can check directly using Kasteleyn orientations), the above F symbol may violate the Majorana fermion parity. So we the state is obstructed if [ω 2 ] is nontrivial.
However, we can introduce a 2D bulk to the above 1D state we are constructing. The difference is that we can use a complex fermion from the 2D bulk to compensate the Majorana fermion parity of the 1D F move Eq. (163). So the new F move reads We have to impose the condition to make the new F move total fermion parity even. So there is no longer fermion parity inconsistency for the 1D ASPT state on the boundary of a 2D bulk. In fact, the 1D F move Eqs. (162) and (164) can be understood as the FSLU transformation U 1 Eq. (160) for a single triangle 012 (see Fig. 7). The upper two links 01 and 12 correspond to the right-hand-side of 1D F move. And the lower link 02 corresponds to the left-hand-side of 1D F move. The additional complex fermion c g0 012 in 1D F move Eq. (164) is merely the decorate complex fermion at the center of the 2D triangle. This picture relates the 1D FSPT obstruction and the 2D FSPT trivialization, and explains the trivialization relation Eq. (165) in a simple way.

F. Classification of 2D FSPT phases
The general classification of 2D FSPT phases is as follow. We first calculate the cohomology groups H 1 (G b , Z 2 ), H 2 (G b , Z 2 ) and H 3 (G b , U (1) T ). For each n 1 ∈ H 1 (G b , Z 2 ), we solve the twisted cocycle equation Eq. (14) for n 2 .
For each solution n 2 , we solve the twisted cocycle equation Eq. (14) for ν 3 . If n 2 and ν 3 are in the trivialization subgroup Γ 2 and Γ 3 in Eq. (15), then they are trivialized by boundary ASPT states. So the obstruction-free and trivialization-free (n 1 , n 2 , ν 3 ) fully classify the 2D FSPT phases.
Similar to the bosonic case, we can also use the 2D FSLU transformations to construct the commuting projector parent Hamiltonians. The procedure is tedious but straightforward. For the case of complex fermion decorations only, it is given explictly in Refs. [72]. The terms of the Hamiltonian are sequences of fermionic F moves that changes the group element label of a vertex from one to another. All the terms commute with each other, because our FSPT wave function is at the fixed-point.

VI. FIXED-POINT WAVE FUNCTION AND CLASSIFICATION OF FSPT STATES IN 3D
The fixed-point wave function for FSPT state in 3D have four layers of degrees of freedom. It is a superposition of all possible basis states as (we omit the 2D p + ip chiral superconductor layer in the figure for simplicity): The basis state is a labelled by group elements of G b on each vertex. On the plane dual to each link, we put 2D p + ip chiral superconductor. Different from other layers, the p + ip superconductor layer does not have fixed-point wave function on discrete lattice [80]. So we will not discuss this layer decoration until at the end of this section. Along the dual link of each triangle, we decorate Kitaev chains (see green lines in the above equation). And at the center of each tetrahedron, we decorate some complex fermions (see blue dots in the above equation).

A. Four layers of degrees of freedom
Similar to 2D, we construct FSPT states on 3D lattice by decorating complex fermions, Kitaev chains and p + ip superconductors to the BSPT states. Therefore, there are four layers of degrees of freedom including the bosonic ones on 3D triangulation lattice: • |G b | level bosonic (spin) state |g i (g i ∈ G b ) on each vertex i.
• |G b | species of complex fermions c σ ijkl (σ ∈ G b ) at the center of each tetrahedron ijkl .
• |G b | species of 2D p + ip chiral superconductors [may have several copies indicated by n 1 (g i , g j ) ∈ Z T ] on the plane dual to link ij . The boundary chiral Majorana modes (along the link dual to some triangle) are ψ σ ij,L;α or ψ σ ij,R;α (σ ∈ G b ) depending on the chirality (left/right-hand-rule with respect to the oriented link ij ). Here α labels the number of the chiral Majorana modes (α = 1, 2, ..., |n 1 (g i , g j )|).
The four layers of degrees of freedom are summarized in one tetrahedron of the 3D triangulation lattice in Fig. 8. The four vertices of the tetrahedron are labelled by g 0 , g 1 , g 2 , g 3 ∈ G b . In Fig. 8a, blue ball is the complex fermion c σ 0123 (σ ∈ G b ) at the center of the tetrahedron. Red dots represent Majorana fermions γ σ ijk,A and γ σ ijk,B (σ ∈ G b ) on the two sides of the triangle ijk . We use the convention that the direction pointing from γ σ ijk,A to γ σ ijk,B is the same as the right-hand rule orientation of the triangle ijk . In Fig. 8b, each green area dual to link ij represents the decorated 2D p + ip superconductors. There are |n 1 (g i , g j )| right-moving or left-moving chiral Majorana modes ψ gi i,R/L;α (α = 1, 2, ..., |n 1 |) along the boundary (red line) of the green area dual to each link ij . As discussed in section III B, the symmetry transformation rules of these degrees of freedom under G b are summarize as follows (g, g i , σ ∈ G b ): U (g)ψ σ ij,L;α U (g) † = (−1) ω2(g,σ)+s1(g) ψ gσ ij,g(L);α .
The bosonic degrees of freedom always form a linear representation of G b (and G f ). On the other hand, the fermion modes support projective representations of G b with coefficient (−1) ω2 , and hence linear representations of G f by Eq. (63).

B. Decorations of fermion layers
In this section, we would construct systematic procedures of decorating Kitaev chains and complex fermions to the bosonic basis state |{g i } . Each layer of the degrees of freedom will twist the consistent equations for the next layer. The decoration rules should respect the symmetry in all layers.
We will focus on the Kitaev chain and complex fermion decoration here, and the decorations of 2D p + ip superconductors will be discussed at the end of this section.

Kitaev chain decoration
The Kitaev chain decoration in 3D is similar to the constructions in the Ref. 53. The difference is that we would put |G b | species of Majorana fermions rather than one. However, we still put only one species of Majorana fermions into nontrivial pairings along the decorated Kitaev chain.
a. Decoration procedure. For a given 3D triangulation lattice, we first construct the resolved dual lattice (red lattice shown in Fig. 9). Our Majorana fermions γ σ ijk,A and γ σ ijk,B (σ ∈ G b ) are at the (red) vertices on the two sides of each (black) triangle ijk . The red arrow is follow the convention that the direction from γ σ ijk,A to γ σ ijk,B is the same as the right-hand rule orientation of the triangle ijk . The direction of vacuum pairing between them is from A to B: −iγ σ ijk,A γ σ ijk,B = 1 when acting on the state. To decorate Kitaev chains on the red lattice, we also should add arrows to the small red tetrahedron inside each black tetrahedron (see Fig. 9). These red arrows are constructed from the discrete spin structures (a choose of trivialization of Stiefel-Whitney homology class w 1 dual to cohomology The Kitaev chain decoration on the red lattice is specified by n 2 (g i , g j , g k ) ∈ Z 2 , which is a function of three group elements g i , g j , g k ∈ G b . If n 2 (g i , g j , g k ) = 0, the Majorana fermions γ σ ijk,A and γ σ ijk,B on the two sides of the triangle ijk are in vacuum pairings: −iγ σ ijk,A γ σ ijk,B = 1 (for all σ ∈ G b ). On the other hand, if n 2 (g i , g j , g k ) = 1, we will decorate a Kitaev chain going though the triangle ijk . For all |G b | species of Majorana fermions, we only put γ gi ijk,A and γ gi ijk,B to be in the nontrivial pairing. And all other |G b | − 1 species of Majorana fermions γ σ ijk,A and γ σ ijk,B with σ = g i are still in vacuum pairings. Here is an example of the decoration of Kitaev chain going though triangles 013 and 023 of the tetrahedron 0123 (we omit the operator labels of Majorana fermions which are in vacuum pairings along the Kitaev chain): The decorated Kitaev chain is indicated by a green line (γ g0 013B -γ g0 013A -γ g1 123A -γ g1 123B ). Along the Kitaev chain, there is a nontrivial pairing between γ g1 123A and γ g0 013A . And the Majorana fermions γ σ 013A/B (σ = g 0 ) and γ σ 123A/B (σ = g 1 ) are all in vacuum pairings. For the triangles without Kitaev chain going though (triangles 012 and 023 ), the Majorana fermions on their two sides are all in vacuum pairings (γ σ 012A/B and γ σ 023A/B for all σ ∈ G b ). In summary, we have the following Majorana fermion pairings: Both the trivial and nontrivial Majorana fermion pairings are indicated by gray ellipses. The blue arrow means that the pairing direction of Majorana fermions may be changed compared to the local Kasteleyn orientation indicated by red arrow. We will discuss more about the detailed pairing directions and why they are symmetric later. b. Consistency condition. According to our decoration rule, the total number of decorated Kitaev chains going though the four boundary triangles of a given tetrahedron 0123 is (dn 2 )(g 0 , g 1 , g 2 , g 3 ) = n 2 (g 1 , g 2 , g 3 ) + n 2 (g 0 , g 2 , g 3 ) + n 2 (g 0 , g 1 , g 3 ) + n 2 (g 0 , g 1 , g 2 ). (176) Since we are constructing gapped state without intrinsic topological order, there should be no dangling free Majorana fermions inside any tetrahedron. So the number of total Kitaev chains going though the boundary of a tetrahedron should be even. We therefore have the (mod 2) equation as the consistency condition for Kitaev chain decorations. c. Symmetric pairing directions. Now let us turn back to the details of symmetric Majorana fermion pairings inside each tetrahedron of the 3D triangulation lattice. Our strategy of constructing G b -symmetric pairings is the same as in the 2D case: we first consider the standard tetrahedron with g 0 = e, and then apply a U (g 0 )-action to obtain all other non-standard tetrahedra with generic group element labels. In this way, the Majorana fermion pairings are automatically symmetric, because of the symmetry transformation rule of the pairing projection operators Eq. (99). For the standard tetrahedron, the Majorana fermions are paired according to the local Kasteleyn orientations indicated by red arrows. And the pairings in the non-standard tetrahedron is obtained by a U (g 0 )-action as follows: On the other hand, if n 2 (g i , g j , g k ) = 1 (there is a Kitaev chain going though this triangle), we will pair the Majorana fermion inside the triangle with another one belonging to another triangle with also n 2 = 1 [for example, γ g1 123A , γ g0 023B , γ g0 013A and γ g0 012B ) inside the tetrahedron 0123 , for we have (dn 2 )(g 0 , g 1 , g 2 , g 3 ) = 0 (mod 2) from Eq. (177). There are in total (4 × 3)/2 = 6 possible nontrivial pairings inside the tetrahedron 0123 (i.e., the six links of the small red tetrahedron inside the big tetrahedron 0123 ). The Majorana pairing projection operators of them are as follows:  182) and (185)]. The first two pairings Eqs. (180) and (181) belongs to neither of the above two cases. So their pairing direction is the same as the red arrow Kasteleyn orientations even after U (g 0 )-action. There is another subtlety when n 2 = 1 for all the four triangles of a tetrahedron. There are four strings of Kitaev chains meeting at the tetrahedron 0123 . In this case, we should resolve the crossing point of the four strings. We use the convention that the Majorana fermions γ g1 123A and γ g0 013A are paired, and γ g0 023B and γ g0 012B are paired (see Fig. 10a). Of course, all other Majorana fermions γ σ ijk,A/B with σ = g i are still in vacuum pairings. This resolvation convention is the same as Ref. 53. d. Majorana fermion parity. Since the symmetry action may change the pairing directions inside a tetrahedron, the Majorana fermion parity of this tetrahedron may also be changed. We can calculate the fermion parity difference between the standard and non-standard tetrahedra by counting the number of pairing arrows that are reversed by U (g 0 )-action. It of course depends on the n 2 configurations. We can use, for example, n 2 (g 0 , g 1 , g 2 )n 2 (g 1 , g 2 , g 3 ) = 0, 1 to indicate whether γ g1 012B and γ g1 123A are paired or not. So the Majorana fermion parity change inside the triangle is in general given by ∆P γ f (0123) = (−1) ω2(g0,g −1 0 g1)n2(g1,g2,g3)+s1(g0)[n2(g0,g2,g3)n2(g0,g1,g2)+n2(g0,g1,g3)n2(g1,g2,g3)] where we have used the higher cup product definition (n 2 1 n 2 )(0123) = n 2 (023)n 2 (012) + n 2 (013)n 2 (123). The above equation is a summary of the phase factors appearing in Eqs. (180)-(185). The first term (−1) ω2 appears iff n 2 (g 1 , g 2 , g 3 ) = 1. And the second term (−1) s1(g0) in the first line of Eq. (186) appears when n 2 (g 0 , g 2 , g 3 )n 2 (g 0 , g 1 , g 2 ) = 1 or n 2 (g 0 , g 1 , g 3 )n 2 (g 1 , g 2 , g 3 ) = 1. Note that different from the 2D case, we can not find a single Majorana fermion involving all the Majorana fermion parity configurations. The above discussions of Majorana fermion pairings also apply to the negative orientated tetrahedra. The red arrow local Kasteleyn orientations inside a tetrahedron is given in Fig. 9b. The symmetric Majorana fermion pairing in a non-standard tetrahedron is also obtained by a U (g 0 )-action from the standard tetrahedron. We also have Majorana pairing projection operators similar to Eqs. (180)-(185) for negative orientated tetrahedron. For n 2 configurations with four Kitaev chains meeting at one negative oriented tetrahedron, we would use the resolvation convention shown in Fig. 10b. And the expression of Majorana fermion parity changes Eq. (186) is valid for both positive and negative oriented tetrahedra.
To sum up, although there are |G b | species of Majorana fermions, we decorate exactly one Kitaev chain to each intersection line of symmetry domain walls specified by n 2 configurations. The decoration is compatible with symmetry actions. The Majorana fermion parity of a tetrahedron is changed according to Eq. (186) compared to the local Kasteleyn oriented pairings.

Complex fermion decoration
The rules of complex fermion decoration are much simpler than that of the Majorana fermions. The decoration is specified by a Z 2 -valued 3-cochain n 3 ∈ C 1 (G b , Z 2 ). If n 3 (g i , g j , g k , g l ) = 0, all the modes of complex fermions c σ ijkl (σ ∈ G b ) at the center of tetrahedron ijkl (i < j < k < l) are unoccupied. On the other hand, if n 3 (g i , g j , g k , g l ) = 1, exactly one complex fermion mode c gi ijkl will be decorated at the center of tetrahedron ijkl (see Fig. 8a). All other complex fermions c σ ijkl (σ = g i ) are still in vacuum states. It is simple to check that the complex fermion decoration is G b -symmetric. Under a U (g)-action, the bosonic vertex labels {g i } becomes {gg i }. And we should decorate c ggi ijkl to the tetrahedron ijkl if n 3 (gg i , gg j , gg k , gg l ) = n 3 (g i , g j , g k , g l ) = 1. This is exactly the complex fermion c gi ijkl by a U (g)-action.

C. F moves
The 3D Pachner move for different triangulations of 3D spacial manifold induces FSLU transformation of the FSPT wave functions on the lattices. Since FSPT state is invertible, we only need to consider one of the many Pachner moves. Other Pachner moves can be derived from this one using the invertibility and unitarity of the move. We can first define the standard F move with g 0 = e, then other non-standard ones can be obtained by simply a U (g 0 )-action. The standard F move is given by on the (black) triangulation lattice, and on the (red) resolved dual lattice. This Pachner move involves five vertices from 0 to 4 , with group element labels e, g −1 0 g 1 , g −1 0 g 2 , g −1 0 g 3 and g −1 0 g 4 in G b . There are three tetrahedra ( 0123 , 0134 and 1234 ) in the right-hand-side figure, and two tetrahedra ( 0124 and 0234 ) on the left-hand-side figure of Eq. (187). The Pachner move removes the link 13 on the right. Note that we omit the p + ip layer in the above figures. And for each tetrahedron, the meeting point of the four red/green strings in Eq. (187) should be resolved to a small tetrahedron with local Kasteleyn orientations in Eq. (188) (using the convention Fig. 9b). The blue arrows of the red links inside the only non-standard tetrahedron 1234 [see the right-hand-side figure of Eq. (188)] mean that their directions may be changed under U (g −1 0 g 1 )-action compared to the red arrow local Kasteleyn orientations [see Eq. (178)]. Green line represents the decorated Kitaev chain specified by the n 2 data. We also omit the blue ball symbols for the decorated complex fermions at the center of the tetrahedra in the figures.
The explicit expression of the standard FSLU F operator for the above Pachner move is We used the abbreviationīj for g −1 i g j in the arguments of F and ν 4 . Different from ν 4 , the 3-cochain n 3 is symmetric under G b -action. So there is no difference between homogeneous and inhomogeneous n 3 . We use n 3 (ijkl) to represents n 3 (g i , g j , g k , g l ) = n 3 (g −1 i g j , g −1 j g k , g −1 k g l ) for short. In the following, we would explain the terms in F operator Eq. (189) one by one. There is some subtlety about the Majorana fermion parity changes for the X operator. And we will give the explicit expression for X in the next subsection.
The U (1) phase factor ν 4 (01,12,23,34) = ν 4 (g −1 0 g 1 , g −1 1 g 2 , g −1 2 g 3 , g −1 3 g 4 ) in the front of F symbol is a inhomogeneous 4-cochain depending on four group elements of G b . By definition, the inhomogeneous cochain is related to the homogeneous one by with the first argument of the homogeneous ν 4 to be the identity element e ∈ G b . As will be discussed below, we can relate ν 4 (e, g −1 0 g 1 , g −1 0 g 2 , g −1 0 g 3 , g −1 0 g 4 ) and ν 4 (g 0 , g 1 , g 2 , g 3 , g 4 ) from the symmetry conditions of F . Different from the special case G f = Z f × G b , they do not equal to each other in general.
The complex fermion term of the form c † c † ccc annihilate three complex fermions at the three tetrahedra of the righthand-side figure and create two on the left-hand-side figure of Eq. (187). Following section VI B 2, the tetrahedron ijkl is decorated by complex fermion c gi ijkl . So in the standard F move, only the last fermion c 1234 has group element label g −1 0 g 1 , and all other four fermions have group element label e. We note that, different from the special case G f = Z f 2 × G b where we have dn 3 = n 2 n 2 [53], the complex fermion number n 3 has more complicated relation with the n 2 data in general.

Majorana fermion parity and X operator
In this subsection, we will give explicit expression for the X operator in the standard F move Eq. (189), which is related to the Kitaev chain decorations.
As In the standard F move Eq. (187), there are in total five relevant tetrahedra. Four of these tetrahedra are standard, with group element label of the first vertex e ∈ G b . However, the tetrahedron 1234 on the right-hand-side of Eq. (187) is non-standard and has first vertex label g −1 0 g 1 . Therefore, only inside the tetrahedron 1234 of the standard F move, the pairing directions of the Majorana fermions may be changed. The pairings are given by the projection operators Eqs. (180)-(185), with the replacement (g 0 , g 1 , Furthermore, as discussed also in section VI B 1 d, the Majorana fermion parity change for a non-standard tetrahedron 0123 compared to the local Kasteleyn orientations is given by Eq. (186). For the tetrahedron 1234 of the standard F move, the Majorana fermion parity change compared to the local Kasteleyn orientations is then This is obtained from Eq. (186) by simply the replacement (g 0 , g 1 , g 2 , g 3 ) → (g −1 0 g 1 , g −1 0 g 2 , g −1 0 g 3 , g −1 0 g 4 ). Combined it with Eq. (191), the total Majorana fermion parity change under the standard F move Eq. (187) is We note that there is no difference between the homogeneous and inhomogeneous notations for Z 2 -valued cocycles s 1 , ω 2 and cochain n 2 , for they are symmetric under G b -action. So we can just use (01234) to represent (g 0 , g 1 , g 2 , g 3 , g 4 ) or (g −1 0 g 1 , g −1 1 g 2 , g −1 2 g 3 , g −1 3 g 4 ) in the above equation. This is very different from the U (1) phase factor ν 4 .
The philosophy of this splitting is as follows. The first part α 4 has the property that α 4 (01234) = 1 implies n 2 (234) = 1, because all of the terms in α 4 (01234) contain the factor n 2 (234). However, the second part β 4 does not contain the factor n 2 (234). For a given Kitaev chain configuration n 2 , if α 4 (01234) = 1 and β 4 (01234) = 0, we can conclude that the Majorana fermion parity of the Kitaev chain going through the link 234 is changed. We can add a Majorana fermion operator γ 234B to the X operator of Eq. (189) to represent the correct fermion parity change. On the other hand, if α 4 (01234) = 0 and β 4 (01234) = 1, the Majorana fermion parity of the Kitaev chain going through the link 123 is changed. We can add another Majorana fermion operator γ 123A to the X operator (we can also use γ as another convention). In this case, adding γ 234B does not make sense, because the Kitaev chain may even do not go through this Majorana fermion.
With the above understanding, the explicit form of X 01234 [n 2 ] in the standard F move Eq. (189) can be expressed by Majorana fermion pairing projection operators as: The two Majorana fermion operators γ where we do not need γ 123A , because the configuration n 2 (134) = 0 implies β 4 (01234) = s 1 (01)n 2 (134)n 2 (123) = 0. The F symbol Eq. (189) should be a FSLU operator. It should be both fermion parity even and symmetric under G b -action. We can use these constraints to obtain several consistency equations for the cochains n 2 , n 3 and ν 4 .

Fermion parity conservation
As discussed in previous subsection, the Majorana fermion parity change for the standard F move is given by Eq. (191), if the pairings are according to the local Kasteleyn orientations. From the symmetry action on the only non-standard tetrahedron 1234 of the standard F move, there is an additional Majorana fermion parity change Eq. (192). By combine them, we obtain the total Majorana fermion parity change Eq. (193) for the standard F move.
The decorations of Majorana fermions and complex fermions are not independent, and should satisfy the above constraint.
We note that, if we consider the special case of unitary symmetry group G f = Z f 2 × G b (i.e., ω 2 = 0 and s 1 = 0), the above equation is reduced to the previous known result dn 3 = n 2 n 2 [53,62,63].

Symmetry condition
In the previous constructions, we only considered the standard F move Eq. (187) with the first vertex label e ∈ G b . The non-standard F move is defined to be obtained from the standard one by a U (g 0 ) symmetry action. In such way, the F moves are symmetric under G b -actions. In this subsection, we will derive the symmetry transformation rules for the F move, and the U (1) phase factor ν 4 in the front of F move.
According to Eq. (65), we have the following commuting diagram for the standard and non-standard F moves (we omit the decorated fermion layers in the figures): So the non-standard F operator is defined as The non-standard F moves constructed in this way are automatically symmetric, because one can show the transformation rule for the non-standard F moves F (gg 0 , gg 1 , gg 2 , gg 3 , gg 4 ) = U (g)F (g 0 , g 1 , g 2 , g 3 , g 4 )U (g) † , using Eq. (62) and the fact that F operator is fermion parity even. Using a U (g 0 )-action on the standard F operator Eq. (189), we can obtain the non-standard F symbol expression as F (g 0 , g 1 , g 2 , g 3 , g 4 ) = ν 4 (g 0 , g 1 , g 2 , g 3 , g 4 ) c g0 † The decorated complex fermions now have group element labels g 0 (for the first four complex fermions) or g 1 (for the last complex fermion). And the X operator is with added Majorana fermion γ g2 234B and γ g1 123A , rather than γ From the decoration rules of Majorana fermions and complex fermions constructed in previous sections, the data n 1 , n 2 and n 3 are invariant under G b symmetry actions. The homogeneous cochain ν 4 in the non-standard F move Eq. (205) is a combination of the inhomogeneous ν 4 in the standard F move Eq. (189) and the ±1 signs which appear from the symmetry action. Therefore, we have the following symmetry conditions for the data n 1 , n 2 , n 3 and ν 4 : n 2 (g 0 , g 1 , g 2 ) = n 2 (e, g −1 The We note that some of the terms above can not be expressed as a cup product form. The above equation can be obtained by straightforward calculation using Eq. The symmetry sign O symm 5 will appear later in the twisted cocycle equation for ν 4 as part of the obstruction function [see Eq. (215)]. In the special case of β 4 = 0, the calculation of O 5 is much simpler than the generic case.

D. Super fusion hexagon and twisted cocycle equations
Just as the 2D F move should satisfy the super pentagon equation of superfusion category, the 3D F move should satisfy a super hexagon equation of superfusion 2-category. One should distinguish it from the hexagon equation of braided tensor category, for the former is in the fusion level rather than the braided level. Our 3D FSPT states constructed correspond to some kind of pointed superfusion 2-categories with to a given symmetry group.
It is enough to merely consider the standard super hexagon equations with first vertex label e ∈ G b as coherence conditions. This is because all other non-standard ones can be obtained from it by a U (g 0 ) symmetry action. The standard super hexagon equation is shown in Fig. 11 for the triangulation lattice and Fig. 12 for the dual lattice. Algebraically, we have the following equation where we again usedīj to denote g −1 i g j . In the above equation, only the last F symbol is non-standard. It can be obtained from the standard one by a U (g −1 0 g 1 ) symmetry action [see Eq. (203)]. . The colored numbers i and j in the subscript of F indicate that the link ij with the same color is added after this F move. Since the group element label of the first vertex is e ∈ G b , all the fermionic F moves are standard except0 1 F 12345 . We used a simpler notation F ijkl = F (e,īj,jk,kl) for the standard F move. We also omit all the fermion layers in the figure.
Standard super fusion hexagon equation. Algebraically, this standard super pentagon condition corresponds to Eq. (213). The colored numbers i and j in the subscript of F indicate that the link ij with the same color is added after this F move. Since the group element label of the first vertex is e ∈ G b , all the fermionic F moves are standard except0 1 F 12345 . We used a simpler notation F ijkl = F (e,īj,jk,kl) for the standard F move. We also omit all the fermion layers in the figure.
Using the explicit expression of the standard F move Eq. (189), we can unfold the standard super hexagon equation Eq. (213). By eliminating all complex fermions and Majorana fermions, we can obtain a twisted cocycle equation for the inhomogeneous 3-cochain ν 3 in the standard F move. In general, the twisted cocycle equation reads where O 5 [n 3 ] is a functional of n 3 only (as well as ω 2 and s 1 parametrizing the given symmetry group). The n 2 dependence of O 5 is though dn 3 by Eq. (201). With nonzero ω 2 and s 1 , the obstruction function O 5 [n 3 ] is more complicated than the special result for unitary G f = Z f 2 × G b [53,62,63]. Similar to the 2D case, the obstruction function O 5 [n 3 ] consists of four terms from general considerations:   (195)]. This is true for s 1 = 0 (unitary G b ) or n 1 = 0. The calculation of generic O γ 5 with nonzero β 4 is much more complicated (but the procedures are the same), so leave it for the future. By adding a coboundary (−1) d(s1 n3+n3 3 dn3) to the obstruction function and shifting ν 4 → ν 4 (−1) s1 n3+n3 3dn3 , we can simplify the above obstruction function to  In this subsection, we would give explicit calculations of the four terms of the obstruction function O 5 [n 3 ] in Eq. (215), with the assumption of β 4 = 0.
The first term O symm 5 [n 3 ] comes from the U (01) symmetry action on F (e, 12,13,14,15) in the last term of Eq. (213). Since the homogeneous ν 4 of the non-standard F move is obtained from the standard one by a symmetry action Eq. (210)], there is a sign difference O symm 5 between them. Using the replacement (g 0 , g 1 , g 2 , g 3 , g 4 where the number i is removed in the argument. The obstruction function O γ 5 coming from Majorana fermions can be calculated from the X operators as The average is taken over the Majorana fermion pairing state of the rightmost figure (with projection operator P 1 ) in Figs. 11 and 12. We also inserted P 1 , which is 1 acting on the rightmost state, at the first and the last places of the operator string in Eq.  Let us consider the third case (the fourth row in Table V) We can choose the second term in the above equation and all other projection operators P i with i = 5 to be 1. Then the obstruction function Eq. (225) can be expressed as = (−i)(−1) dn3(1)+ω2(02, 23) .
In such way, Eq. (230) contains all Majorana fermions even times. After reordering these Majorana fermion operators, they all square to one. Finally we obtain Eq. (231) as the obstruction function for this case.
We can similarly calculate O γ 5 [dn 3 ] for all the 8 cases of Majorana fermion parity changes. The information we need in the calculation is shown in Table V. Sometimes, we need not only P 5 but also P 4 . And in P 4 and P 5 , we may need to many Majorana pairing projection operators Eq. (98). They are shown in the fourth and fifth columns of Note that the dn 3 terms in the exponent of (±i) should be understood as taking mod 2 values (can only be 0 or 1). They correspond to the third and sixth cases in Table V. The term (−1) dn3(1) appears in the last column of Table V , the Majorana obstruction function Eq. (232) will be reduced to an expression as a functional of n 2 2 (because of dn 3 = n 2 n 2 in this case). Although the expression is different from the the result in Ref. 53 in appearance, one can show that they are exactly the same after some calculations. It is also the same as the (3+1)D spin cobordism result [62,63].

E. Boundary ASPT states
In the above discussions in this section, we constructed 3D FSPT states by decorating several layers of fermion modes to the BSPT states. The decorations of Kitaev chains and complex fermions are specified by two Z 2 -valued cochains n 2 and n 3 .
In this section, we will show that some of the n 2 and n 3 data in fact correspond to FSPT states belonging to the trivial FSPT phase. This data trivialization can be understood by investigating the boundary ASPT states on the 2D boundary of the 3D FSPT bulk.
1. Boundary ASPT states in Γ 2 with p + ip superconductors Similar to the 2D case, we will show in this section that the 3D FSPT data n 2 = ω 2 is trivialized by boundary ASPT state. The 2D boundary is in fact a state with (one layer) p + ip chiral superconductor decorations.
Since there is no fixed-point wave function construction for 2D p + ip superconductor on discrete lattice [80], its decoration is very different from the Kitaev chains and complex fermions. In the boundary ASPT state construction, we put p + ip superconductor domains around each vertex i of the triangulation lattice, and use symmetric mass terms to gap out the edge modes and glue them together. In the following, we first discuss the procedure of decorating p + ip superconductors. Then we propose a field theory description of symmetrically gapping out the chiral Majorana modes along the edge.
The detailed decorations are as follows. For an arbitrary triangulation of the 2D boundary of a 3D FSPT state, we put a p + ip superconductors on a disk centered at each vertex i. The 2D disks of the superconductors are always gapped. And on the boundary, there are right-moving chiral Majorana modes [see the green arrows in Eq. (233)]. We denote these chiral Majorana modes by ψ gi i,R . A picture of the chiral Majorana modes in a triangle of the 2D boundary of 3D FSPT bulk is The decorated state should be symmetric under G b -action. If the chiral Majorana modes around vertex i with g i = e are ψ e i,R , then for generic g i ∈ G b , the decorated p + ip superconductors should have chiral Majorana modes ψ gi i,gi(R);α according to the symmetry transformation rules: Here, we denote g i (R) = R if s 1 (g i ) = 0 (g i is unitary), and g i (R) = L if s 1 (g i ) = 1 (g i is antiunitary). The physical meaning is that the unitary symmetry U (g) only changes the group element label of chiral modes from g i to gg i . However, time reversal symmetry would transform p + ip superconductors to p − ip superconductors, and vice versa. So the right-moving and left-moving chiral Majorana modes on the edge are switched. The next step in our construction is to gap out all the chiral Majorana modes to obtain a fully gapped boundary state. Consider the interface between two p + ip domains labelled by g 0 and g 1 shown below There is a necessary condition for gapping out the chiral Majorana modes ψ g0 0,R/L and ψ g1 1,R/L shown by green arrows above: the number of net chiral modes along the direction dual to 01 is zero. If s 1 (g −1 0 g 1 ) = 1, then one of the two right-moving chiral Majorana modes is reversed. There are two chiral modes moving in the same direction along the link dual to 01 . This is illegal for we want the boundary to be a gapped state. So we conclude the boundary ASPT state with p + ip superconductors is impossible if there are some antiunitary symmetries in G b . In the following discussions of ASPT state with p + ip superconductors, we will always assume G b is unitary.
Let us introduce explicitly the symmetric mass terms to gap out the chiral Majorana modes. Similar to the previous discussions, we can first assume the group element labels of link 01 are e and g −1 0 g 1 . The standard mass terms along the edge dual to link 01 is The non-standard mass terms are obtained from the above standard ones by a symmetry action: where we used the symmetry transformation rule Eq. (234) of Majorana modes. Note that there is no minus sign from the imaginary unit i, for G b should be unitary according to the previous discussions. The mass terms constructed above are of course G b -symmetric, since they are all obtained from the standard mass terms by a symmetry action. It is well known that changing the sign of the mass m of two counter-propagating chiral Majorana modes would induce a phase transition from a non-topological superconductor to a topological superconductor [81]. If the resulting gapped edge is a 1D topological superconductor, there is effectively a Kitaev chain going along the direction of the chiral modes [see the dashed red line in Eq. (236)]. We can assume the standard mass terms Eq. (237) with m > 0 correspond to the trivial gapped phase. Then the true mass terms Eq. (238) induce effective Kitaev chains going though the link 01 . The number of effective Kitaev chains equals to the number of negative mass terms: where we have used (dω 2 )(g 0 , g −1 0 g 1 , g −1 1 g 2 ) = 0 (mod 2). If ω 2 is nontrivial in Eq. (240), there may be odd number of Kitaev chains going into the triangle 012 . Since our state is on the boundary of a 3D FSPT bulk, we can connect the Kitaev chain to the 3D bulk FSPT state [see Eq. (173)]. The number of Kitaev chains going though a triangle of 3D FSPT state is exactly the n 2 data which is discussed in detail in section VI B 1. So we have n 2 = ω 2 to have a gapped state (including both the boundary and the bulk). Since we have constructed a gapped symmetric (ASPT) state without topological order on the boundary of the 3D FSPT state, we conclude that the bulk FSPT with n 2 = ω 2 is trivial. This is the origin of the trivialization group for 3D FSPT phases claimed in Eq. (19). We note that p + ip superconductor is incompatible with time reversal symmetry. So there is no Γ 2 trivialization if G b contains anti-unitary symmetry. In other words, Eq. (241) is equivalent to because 2. Boundary ASPT states in Γ 3 with p + ip superconductors The ASPT states in Eq. (241) is realized as one layer of 2D p + ip superconductor [n 0 = 1 ∈ H 0 (G b , Z T )] on the boundary of 3D FSPT states with n 2 = ω 2 Majorana chain decorations. The fluctuating Majorana chains (the n 2 data) in the 3D bulk become the effective Majorana chains along the 1D domain walls of 2D boundary p + ip superconductors. Therefore, there is a gapped symmetric boundary without topological orders and the 3D bulk is trivialized.
The ASPT state considered in this section has two layers of 2D p + ip superconductors [n 0 = 2 ∈ H 0 (G b , Z T )]. So according to Eq. (241), there is no Γ 2 trivialization. However, we will show below that there is a Γ 3 trivialization of 3D FSPT state with n 3 = ω 2 1 ω 2 complex fermion decorations. This is related to the fact that the F move of n 0 = 2 p + ip superconductors on the 2D boundary changes fermion parity by ∆P ψ f (F ) = (−1) ω2 1ω2 . This is similar to the Γ 3 trivialization related to n 1 (2D boundary Majorana chain decorations) discussed in the next section.
The setup of n 0 = 2 ASPT state is similar to the state discussed in Section VI E 1. Near the vertex i of the space triangulation, we put a disk of two layers of p + ip superconductors labelled by g i . Along the boundary, we have two chiral Majorana modes indicated by green curves in, for example, Eq. (233) and Eq. (236). After adding symmetric mass terms along the line dual to link ij , we can gap out the chiral Majorana modes. If the sign of mass (−1) ω2(gi,g −1 i gj ) [see the discussions above Eq. (239)] is negative, there are n 0 = 2 effective Majorana chains going though the link ij . Since we want to analyze the fermion parities, we have to pair up the Majorana fermions in the effective Majorana chains. If ω 2 (g i , g −1 i g j ) = 1, there are n 0 = 2 effective Majorana chains, leaving two Majorana fermions on each side of link ij . We can pair them up as shown in Fig. 13b (always from α = 1 to α = 2 on each side). Note that the vacuum pairings for ω 2 (g i , g −1 i g j ) = 0 are the standard A to B pairings (see Fig. 13a). So the fermion parity of the nontrivial pairings is always odd (compared to the vacuum pairings), as the small loop with length four is non-Kasteleyn-oriented. . We note that α = 1, 2 effective Majorana chains should be understood as the stacking in the direction perpendicular to the paper. They are not related to the branching structure (black arrows). Now we can consider the p + ip superconductor configurations in one triangle 012 . The three mass term signs ω 2 (g i , g −1 i g j ) for the three links are independent to each other. So we have in total 2 3 = 8 configurations, which can be divided into four cases. (1) All the three ω 2 's are one. Then all the effective Majorana fermions near the three links are in the vacuum pairings shown in Fig. 13a. And the fermion parity of this configuration is P ψ f = +1. (2) One of the three ω 2 's is one (see Fig. 14a of ω 2 (g 1 , g −1 1 g 2 ) = 1 for example). Then there is only one length-4 loop with non-Kasteleyn-orientation. And the fermion parity is P ψ f = (−1) 1 = −1.
If the exponent in Eq. (244) is a nontrivial cocycle in H 3 (G b , Z 2 ), then the 2D F move necessarily breaks the fermion parity of the p+ip superconductor system. However, if we introduce a 3D FSPT bulk with complex fermion decoration n 3 = ω 2 1 ω 2 , the total fermion parity will be preserved. This is simply because the 2D boundary F move will also change the bulk complex fermion number by n 3 (g 0 , g 1 , g 2 , g 3 ). Therefore, two layers of p + ip superconductors can be viewed as a 2D ASPT state which trivializes the 3D FSPT state with complex fermion decoration n 3 = ω 2 1 ω 2 . So we have a trivialization group Γ 3 as Note that we introduced n 0 ∈ H 0 (G b , Z T ) in the above expression. Only unitary G b would be compatible with p + ip chiral superconductors. And odd n 0 (with nontrivial ω 2 ) already leads to nontrivial trivialization group Γ 2 Eq. (241) in a lower level. Only n 0 = 2 will produce nontrivial Γ 3 and a nontrivial ASPT state on the boundary of 3D FSPT with n 3 = ω 2 1 ω 2 . An explicit example of this Γ 3 trivialization Eq. (245) with n 0 = 2 is the FSPT state with

Boundary ASPT states in Γ 3 with Kitaev chains
There is another layer of boundary ASPT state for the 3D FSPT state. This boundary ASPT state has Kitaev chain decorations, and trivialize the complex fermion decoration data for some n 1 ∈ H 1 (G b , Z 2 ). The second part s 1 n 1 n 1 is discussed in detail in Ref. 66. It is used to construct a 2D ASPT state on the boundary of 3D FSPT state with time reversal symmetry T 2 = 1.
The generic boundary ASPT state with Kitaev chains can be constructed similar to the 2D FSPT state with Kitaev chain decorations in section V B 1. We put (at most) one Kitaev chain along the (red) dual link on the 2D boundary according to the decoration data n 1 ∈ H 1 (G b , Z 2 ). The fermion parity change of the Majorana fermions under the boundary 2D F move is again given by [see Eq. (117)]: If the exponent of (−1) is not a Z 2 -valued coboundary, we can not preserve the boundary fermion parity even introduce complex fermion decorations on the boundary. However, since the ASPT state is on the boundary of a 3D FSPT bulk, we can simply choose the bulk complex fermion decoration data n 3 with Eq. (246). Therefore, the total system (including both the boundary and the bulk) has definite total fermion parity.
Since we construct a gapped symmetric ASPT state without topological order on the boundary of the 3D FSPT state, we conclude that the bulk FSPT with n 3 = ω 2 n 1 + s 1 n 1 n 1 is trivialized. This is the origin of the n 1 part of the trivialization group Γ 3 : Combine it with Eq. (245), we obtain the trivialization group which is claimed in Eq. (19).
F. An additional layer of p + ip SC decorations Apart from the Kitaev chain and complex fermion layers, there is an additional layer of 2D p + ip chiral superconductor (SC) decorations specified by n 1 ∈ H 1 (G b , Z T ) for 3D FSPT states. The decoration of this layer is possible only when G b is not unitary. Since there is no fixed-point wave function constructions of chiral states on discrete lattice [80], we discuss it after all the other layers in this section. However, we can put a 2D continuum (infinite number of degrees of freedom) free fermion p + ip SC state on the decoration plane with the bulk mass approaching positive infinity. As will be shown below, the obstruction function for the p + ip chiral superconductor decoration is ω 2 n 1 + s 1 n 1 n 1 . For the 3D topological superconductor with time reversal symmetry T 2 = −1, the obstruction function equals to zero identically. So this p + ip layer will not twist the obstruction functions of other higher layers. In this way, we can fully classify 3D T 2 = −1 topological superconductors.

Consistency condition
We use n 1 (g i , g j ) ∈ Z to indicate the number of decorated p + ip chiral superconductor layers on the plane dual to link ij (see Fig. 15). If n 1 (g i , j ) < 0, we would decorate inverse p + ip, i.e. p − ip, chiral superconductors. So the number of chiral Majorana modes on the boundary of the plane is |n 1 (g i , g j )|. The direction of the chiral Majorana modes (see red arc arrows in Fig. 15) form right-hand (left-hand) rule with respect to the oriented link ij if n 1 (g i , j ) > 0 (n 1 (g i , j ) < 0). For a triangle ijk , there are three p + ip superconductor planes intersecting at the link dual to the triangle (see the red link in Fig. 15). Since we are constructing a gapped state, there should be no chiral Majorana mode along this link. So we have the gappable condition: (dn 1 )(g 0 , g 1 , g 2 ) = g0 n 1 (g 1 , g 2 ) − n 1 (g 0 , g 2 ) + n 1 (g 0 , g 1 ) = 0, which merely states that the number of left-moving and right-moving chiral Majorana modes along the (red) link dual to triangle 012 equal to each other. Note that the first term in Eq. (250) has a g 0 symmetry action, which adds a minus sign if s 1 (g 0 ) = 1. The physical meaning is that the time reversal symmetry would change n 1 to −n 1 , and reverse the direction of the boundary chiral Majorana modes. If g ∈ G b is unitary, then one can show n 1 (g k ) = kn 1 (g) for all k ∈ Z from Eq. (250). Since we consider only finite G b , we must have n 1 (g) = 0 for all unitary g ∈ G b . So

Symmetry transformations
The chiral Majorana modes along the boundary of the plane dual to link ij will be denoted by ψ gi ij,R;α or ψ gi ij,L;α (α = 1, 2, ..., |n 1 (ij)|) for n 1 (ij) > 0 or n 1 (ij) < 0, respectively. As discussed in VI A, the symmetry transformation rules for the chiral Majorana modes are given by Eqs. (171) and (172). The ω 2 part is the usual projective representation factor for G b (such that it is a linear representation of G f ). The s 1 part is more subtle, which needs some explanations.
To be more specific, the time reversal symmetry acts on the chiral Majorana modes as Basically, it changes ψ R to ψ L , and ψ L to −ψ R (forgetting the ω 2 factor). So the left-moving/right-moving chiral Majorana modes form time reversal Kramers doublet with T 2 = −1. The reason to choose this symmetry transformation convention is as follows. If we fold the three (green) planes in Fig. 15 into one plane, we should obtain a symmetric state on the pure 2D plane. The system consists of several copies of p + ip and p − ip chiral superconductors. However, it is known that the we only have nontrivial 2D topological superconductors for time reversal symmetry T 2 = −1 [76]. The T 2 = 1 case corresponds to the ASPT state which can only exist on the boundary of a 3D bulk [66] (see also the FSPT classification examples in section E 2). Therefore, we choose the chiral Majorana modes to form Kramers doublet under time reversal symmetry.

Symmetric mass terms
To obtain a symmetric gapped state, we should add symmetric mass terms to the three kinds of chiral Majorana modes along the link dual to each triangle ijk . The strategy is again first considering the standard mass term for the standard triangle with first vertex label e ∈ G b . The standard mass term will have the following form where at least one of the two chiral Majorana modes are labelled by the identity element e of G b . The subscript "out" and "in" indicate the actual direction of the chiral Majorana modes along the (red) link dual to the triangle, i.e., going outside or inside the triangle (using right-hand rule). This direction is not locked with the sign of n 1 , for the three links of the triangle have different right-hand rule direction (see the red arc arrows in Fig. 15). We always put out-going mode in front of in-going mode. This is because we have to set a rule to know whether there is an effective Kitaev chain compared to the "trivial" gapped edge by checking simply the sign of the mass. Other non-standard mass terms are obtained from the standard one by a symmetry action. There will be additional minus sign for the mass from the symmetry action. The negative mass will induce effective Kitaev chain along the gapped edge (see Fig. 15). In the following, we will consider separately the symmetric mass terms for nontrivial ω 2 and s 1 .
We first consider the standard mass terms for the standard triangle 012 with group element labels e, g −1 0 g 1 and g −1 0 g 2 for three vertices. According to Eq. (250), the number of left-moving and right-moving modes along the (red) link dual to the triangle should be the same. For simplicity, we assume all of n 1 (01) = n 1 (e, g −1 0 g 1 ), n 1 (12) = n 1 (g −1 0 g 1 , g −1 0 g 2 ) and n 1 (02) = n 1 (e, g −1 0 g 2 ) are positive integers, and satisfy n 1 (02) = n 1 (01) + n 1 (12). Then we need two types of standard mass terms to gap out these modes (assume m > 0): We note that the chiral modes ψ e 01,R;α and ψ g −1 0 g1 12,R;α are going outside the triangle, while ψ e 02,R;α is inside (see the red arc arrows in Fig. 15). So it satisfies the standard mass term rule Eq. (254).
For the non-standard triangle with vertex label g 0 , g 1 and g 2 , we should use a symmetry action to obtain the mass terms: im dxψ g0 01,R;α (x)ψ g0 02,R;α (x) + n1 (12) α=1 (−1) ω2(g0,g −1 0 g1) im dxψ g1 12,R;α (x)ψ g0 02,R;α (x), where the sign (−1) ω2(g0,g −1 0 g1) of the second type mass term comes from the symmetry transformation rule for ψ g −1 0 g1 12,R;α . The total number of effective Kitaev chain going though the triangle 012 is then which is just the number of negative mass terms in Eq. (256). If the three n 1 are not all positive, the mass terms are different from Eq. (256). In general, there are three types of mass terms which gap out different pairs of chiral Majorana modes. The standard form of these three types are where we defined N = max (|n 1 (01)|, |n 1 (12)|, |n 1 (02)|). Note that (at most) only two types of mass terms appear for a given n 1 configuration. We should choose the mass terms which involve the chiral modes corresponding to the biggest |n 1 |. For example, the two types of mass terms in Eq. (255) both involve ψ 02 , because n 1 (02) > 0 is the biggest and N − |n 1 (02)| = 0 for the summation in Eq. (260). The order of the two Majorana modes in a mass term should follow the rule Eq. (254) of the out-going mode appears in the front. It depends on the configuration of n 1 .
The above three equations are only one example. (We also omit the R/L label of the chiral modes for the sign of n 1 is indefinite.) Independent of the signs of n 1 , the effective Kitaev chain number is always Eq. (257). The minus signs of the masses all come from the symmetry sign (−1) ω2(g0,g −1 0 g1) of ψ g −1 0 g1 12,R/L;α . Independent of the signs of n 1 , there are always |n 1 (12)| mass terms associated with this chiral modes [see Eqs. (259) and (260)]. Under a U (g 0 )-action from the standard mass term to the actual mass term, all the terms in Eqs. (259) and (260) have negative masses. So the (mod 2) number of effective Kitaev chains is always Eq. (257).
There are additional signs for the mass terms related to time reversal symmetry. In general, our standard mass term has the form (we omit the group element labels) im dxψ out,s (x)ψ in,s (x), where the out-going mode (going up in Fig. 15) is in front of the in-going mode (going down in Fig. 15). The labels s and s denote R or L, depending on n 1 > 0 or n 1 < 0. Note that the out-going/in-going is not locked with R/L (see the red arc arrows in Fig. 15). Under a unitary symmetry action U (g 0 ), the mass term is transformed to where the sign (−1) ω2 is exactly the transformation sign of ψ g −1 0 g1 discussed previously. So we can obtain the effective Kitaev chain number Eq. (257) in a simple way.
On the other hand, if the symmetry U (g 0 ) is antiunitary, it will reverse the directions of all the chiral modes. The action on the standard mass term Eq.
Apart from the ω 2 term, there are three additional signs. One minus sign comes from the antiunitary action on the imaginary unit i. The second is the sign of switching two chiral modes ψ and ψ . This is because time reversal changes the out-going modes to in-going modes, and vice versa. According to the rule Eq. (254), we should change their orders. The third sign (−1) 1−δ s,s appears only when s and s are different, i.e., the two modes are of different R/L types. This is a consequence of the symmetry transformation rules Eqs. (252) and (253): R → L and L → −R. The only mass term between R and L type modes is Eq. (260) for ψ 01,R/L and ψ 01,L/R . Using the "canonical" 1-cocycle Eq. (251), we conclude that the only mass term with (time reversal related) minus sign is −imψ g1 12 ψ g0 01 with antiunitary g 0 . This case corresponds to s 1 (g 0 ) = 1 − s 1 (g 1 ) = s 1 (g 2 ) = 1 and n 1 (12) = −n 1 (01) = 1 − n 1 (02) = 1. We can summarize this (time reversal related) sign for mass term by (−1) s1(g0)n1(01)n1 (12) . So the (mod 2) number of effective Kitaev chain going though the triangle related to time reversal symmetry is s 1 (g 0 )n 1 (g −1 0 g 1 )n 1 (g −1 In summary, we can use symmetric mass terms to gap out all the chiral Majorana modes along the (red) link dual to the triangle 012 . There are effective Kitaev chains left along the link, with (mod 2) number (ω 2 n 1 + s 1 n 1 n 1 )(g 0 , g −1 0 g 1 , g −1 1 g 2 ) = ω 2 (g 0 , g −1 0 g 1 )n 1 (g −1 1 g 2 ) + s 1 (g 0 )n 1 (g −1 0 g 1 )n 1 (g −1 which is a combination of Eqs. (257) and (264).

Obstruction function
If we consider a tetrahedron 0123 of the 3D triangulation lattice of the spacial manifold, there are four triangles on the boundary and four (red) links with chiral Majorana modes (see Fig. 8b). We should add mass terms H Using the cocycle equations for ω 2 , s 1 and n 1 , we have d(ω 2 n 1 + s 1 n 1 n 1 )(g 0 , g −1 0 g 1 , g −1 1 g 2 , g −1 2 g 3 ) = 0. So total (mod 2) number of effective Kitaev chains for the tetrahedron is (ω 2 n 1 + s 1 n 1 n 1 )(g −1 0 g 1 , g −1 1 g 2 , g −1 2 g 3 ). (266) To make sure that there are no dangling Majorana fermion inside any tetrahedron of the lattice, the number of effective Kitaev chains should equal to the number of decorated Kitaev chains specified by n 2 . So we have the consistency equation dn 2 = ω 2 n 1 + s 1 n 1 n 1 .
If the right-hand-side of the above equation is not a Z 2 -valued 3-coboundary, there is no solution for n 2 on the left-hand-side. We note that the above equation is the same as the 1D consistency equation Eq. (119), although the physical meanings are totally different.

G. Classification of 3D FSPT phases
The general classification of 3D FSPT phases is as follow. We first calculate the cohomology groups H 1 (G b , Z T ), H 2 (G b , Z 2 ), H 3 (G b , Z 2 ) and H 4 (G b , U (1) T ). For each n 1 ∈ H 1 (G b , Z T ), we solve the twisted cocycle equation Eq. (18) for n 2 . For each solution n 2 , we solve the twisted cocycle equation Eq. (18) for n 3 . And for each solution n 3 , we solve the twisted cocycle equation Eq. (18) for ν 4 . If n 2 , n 3 and ν 4 are in the trivialization subgroup Γ 2 , Γ 2 and Γ 3 in Eq. (19), then they are trivialized by boundary ASPT states. So the obstruction-free and trivialization-free (n 1 , n 2 , n 3 , ν 4 ) fully classify the 3D FSPT phases.
We can also use the 3D FSLU transformations to construct the commuting projector parent Hamiltonians. The procedure is again tedious but straightforward. Each term of the Hamiltonian is a sequence of 3D fermionic F moves that changes the group element label of a vertex from g * to g * . Different terms commute with each other for the 3D FSPT wave function is at the fixed-point.

VII. CONCLUSIONS AND DISCUSSIONS
In this paper, we construct gapped fermionic state with symmetry G f by decorating fermionic degrees of freesom. In d spacial dimension, they are constructed using several layers of data (..., n d−1 , n d , ν d+1 ), which is an element in ... × C d−1 (G b , Z 2 ) × C d (G b , Z 2 ) × C d+1 (G b , U (1) T ) (see Table I). There are several consistency conditions for them. Basically, the coboundary of one layer data should equal to a functional of the data of lower layers (see Table II). We can summarize them as a system of twisted cocycle equations: There are two related questions about the constructed FSPT states. The first is that whether the states with different (..., n d−1 , n d , ν d+1 ) data represent distinct FSPT phases. If we have a path of FSLU transformations to connect them, they are in fact in the same phase. So the FSPT classification data should quotient these cases. The second question is what happens to the state that is obstructed by some nontrivial cocycle O i . There are some physical inconsistencies for these states, because all the obstruction functions have physical meanings such as fermion parity conservation (see the last column of Table II). But is it possible to construct such state on the boundary of a one higher dimensional state? If it is possible, we need to understand the physical properties of the bulk, such as whether it is long range entangled or short range entangled.
The answers of the the above two questions are related to the concept of anomalous SPT states [66]. If one of the obstruction functions on the right-hand-side of Eq. (268) is not a coboundary, the state will be obstructed for it violates some physical consistency constraints (see the last column of Table II). However, this state can exist as ASPT state on the boundary of an FSPT state in (d + 1) spacial dimension. The ASPT states in (d − 1) spacial dimensions induces new kinds of coboundaries for the classification data of FSPT phases in d dimensions [52,53,66]. Formally, we can write the new coboundaries as In the first line of the above equation, we identify the FSPT state with (..., n d−1 , n d , ν d+1 ) with another state by stacking the coboundary of one lower dimension state with data (..., n d−2 , n d−1 , ν d ). The coboundary of the latter data, by Eq. (268), are exactly the obstruction functions for the FSPT in one lower dimensions. Therefore, the lower dimensional FSPT obstruction functions will trivialize the higher dimensional FSPT data. Mathematically, the obstruction functions O i for FSPT states in (d − 1) spacial dimensions form a subgroup of the cohomology group H i (G b , Z 2 ) [or H i (G b , U (1) T )]: If O i [n i−2 ] is a nontrivial cocycle in Γ i , the (d − 1) dimensional state with classification data n i−2 is obstructed. On the other hand, if the classification data n i (or ν i ) for d dimensional FSPT state belongs to the subgroup Γ i , it will be trivialized because of the boundary ASPT state. Therefore, the distinct classification data for d dimensional FSPT state is in fact n i ∈ C i (G b , ·)/B i (G b , ·)/Γ i (the coefficient is Z 2 or U (1) T ). For each solution of Eq. (268), we can use the classification data to construct an FSPT state by decorating several layers of fermionic degrees of freedom to the BSPT state. These state belong to different FSPT phases if the data are different in C i (G b , ·)/B i (G b , ·)/Γ i . We can also use the FSPT moves to construct commuting projector parent Hamiltonians. The Hamiltonian consists of local operator which corresponds to a sequence of F moves that changes one vertex label from g * to g * . Different terms commute with each other, because the F moves satisfy the coherence conditions.
We conjecture that in principle our classification scheme for FSPT phases can also be applied to point/space group symmetry, so long as the crystalline principle and spin statistics relations [82] are carefully considered. For example, the mirror symmetry with σ 2 = 1 (Z f 2 × Z P 2 ) should be regarded as a time reversal symmetry with T 2 = P f (Z T f 4 = Z f 2 × ω2 Z T 2 ), while the mirror symmetry with σ 2 = P f (Z T P f 4 = Z f 2 × ω2 Z P 2 ) should be regarded as a time reversal symmetry with T 2 = 1 (Z f 2 × Z T 2 ) [83]. The full details for the classification and construction of point/space group protected FSPT phases will be presented in our future work. Moreover, we also believe that our construction and classification scheme can be applied for continuum Lie group symmetry by using the Borel cohomology. However, it is very difficult to compute the obstruction functions for general Lie group, and we will develop special tools to handle this problem in the future. Finally, how to generalize our framework into FSPT phases protected by super symmetry (SUSY) will be an extremely interesting future direction! The symmetry transformation of the basis state under G f is U P n f g |σ = (−1) n |gσ .
The fixed-point wave function is also a superposition of all basis states with odd fermion parity. One can check the one dimensional representation of G f on this fixed-point wave function: U P n f g |Ψ = (−1) n ν 1 (g)|Ψ . To decorate Majorana fermions on the 2D or 3D triangulation lattices, we need (local) Kasteleyn orientations to design the pairing directions between the Majorana fermions. In the following, we will discuss briefly how to construct (local) Kasteleyn orientations from arbitrary triangulation of the spacial manifold. We refer the interested readers to Ref. 53 for more details.
The (local) Kasteleyn orientations are constructed for arbitrary triangulation T of a spin manifold (spacial manifold with dimension d) with vanishing second Stiefel-Whitney cohomology class [w 2 ] = [0]. Using the mathematical result of Ref. 84, we know the representative on arbitrary triangulation of the Stiefel-Whitney homology class w d−2 , which is the Poincaré dual of w 2 . Then we can construct a resolved dual latticeP with (local) Kasteleyn orientated links.
The procedure of constructing (local) Kasteleyn orientations is as follows [53]: 1. Given a (black) triangulation lattice T with branching structure for a d-dimensional spacial spin manifold; equations shown in Eq. (14) (see also Table II). Since we do not consider the invertible topological order p + ip superconductors as FSPT states, we need only to calculate the obstructions for n 1 and n 2 . We will use frequently the Z 2 and U (1)-valued cocycle invariants in Ref. 16,85, and 86 to check whether a cocycle is trivial or not for finite Abelian groups.
(1) Obstructions for n 1 . From Eq. (14), the obstruction function for n 1 is O 3 [n 1 ] = ω 2 n 1 because of s 1 = 0. If we choose n 1 = n (i) 1 (0 ≤ i ≤ K), the obstruction function is It is known that for finite Abelian groups that the above equation is always a nontrivial cocycle if both n which is a cocycle in H 4 (G b , U (1) T ). Note that if N 0 is odd, we can choose ω 2 = 0, and the equation for n 2 is always dn 2 = 0. So many terms in Eq. (136) vanish, including the (±i) terms. Since there are many different types of n 2 in Eq. (E4), we will discuss them seperately.
Combining the trivializations of n 2 with obstruction-free n 2 number Eq. (E14), the number of trivialization-free obstruction-free n 2 is #(trivialization-free obstruction-free n 2 ) = 1≤i≤K gcd(2, N i ) 1≤i<j≤K gcd 2, N ij N 0ij . (E15) (2) Trivializations of ν 3 . For the bosonic U (1) phase factor ν 3 , the trivialized subgroup in Eq. (15) can be calculated as For the cocycles in the subgroup Z N0 × 1≤i≤K Z 0i of H 3 (G b , U (1) T ) in Eq. (E5), they have a "canonical" form expressed as lower dimensional Z 2 -valued cocycles n 1 . Comparing it with the above equation, we see that if N i is odd, or N i /2 is a integral multiplier of N 0i , then the part of Γ 2 related to Z Ni ⊂ G b (0 ≤ i ≤ K) is trivial in H 3 (G b , U (1) T ). Otherwise, the 3-cocycle ν 3 in Eq. (E17) with k = N i /2 (mod N 0i ) is trivialized by 2D ASPT state. The results calculated above can also be obtained from calculating the cocycle invariants for ν 3 [16,85,86].
In summary, (1) if N 0 is even, one nontrivial ν 3 in Z N0 ⊂ H 3 (G b , U (1) T ) is trivialized. Otherwise, all elements in Z N0 ⊂ H 3 (G b , U (1) T ) are nontrivial. (2) For 1 ≤ i ≤ K, and the subgroup Z N0i ⊂ H 3 (G b , U (1) T ), we also have two possibilities. If N i is even and N i /N 0i is odd [87], then one nontrivial ν 3 in Z N0i ⊂ H 3 (G b , U (1) T ) is trivialized. Otherwise, all elements in Z N0i ⊂ H 3 (G b , U (1) T ) are nontrivial. So the number of trivialization-free cocycles in the The total number of ν 3 that is not trivialized in Eq. (E5) is c. Full classification From the above calculations of obstructions and trivializations, we can obtain the number of 2D FSPT phases with symmetry group Eq. (E1) by combining Eqs. (E11), (E15) and (E19): If N 0 is even, the above equation is reduced to On the other hand, if N 0 is odd, we have After some calculations [88], one can show that the above two equations for the number of 2D FSPT phases are exactly the same as Eqs. (42) and (43) of Ref. 13, which are obtained from a totally different approach of the braiding statistics data of the gauge flux. Furthermore, one can show that the group structure of general group supercohomology classification [89] also agrees with Ref. 13. We note again that we use a different convention of N 0 definition compared to Ref. 13.