Tenfold Topology of Crystals

The celebrated tenfold-way of Altland-Zirnbauer symmetry classes discern any quantum system by its pattern of non-spatial symmetries. It lays at the core of the periodic table of topological insulators and superconductors which provided a complete classification of weakly-interacting electrons' non-crystalline topological phases for all symmetry classes. Over recent years, a plethora of topological phenomena with diverse surface states has been discovered in crystalline materials. In this paper, we obtain an exhaustive classification of topologically distinct groundstates as well as topological phases with anomalous surface states of crystalline topological insulators and superconductors for key space-groups, layer-groups, and rod-groups. This is done in a unified manner for the full tenfold-way of Altland-Zirnbauer non-spatial symmetry classes. We establish a comprehensive paradigm that harnesses the modern mathematical framework of equivariant spectra; it allows us to obtain results applicable to generic topological classification problems. In particular, this paradigm provides efficient computational tools that enable an inherently unified treatment of the full tenfold-way.

The tenfold-way and the mutual relations amongst symmetry classes within it are thus at the physical foundation of our understanding of topological quantum phenomena. One of the insightful ways to derive and decipher the structure of the periodic table of TISC is to naturally encapsulate the tenfold-way within "complex K-theory" and "real K-theory", the former of which is sometimes referred to as 'the simplest generalized cohomology theory'. Since our understanding of their importance, this and other cohomology theories have been used throughout physics to explore and understand various phenomena. Examples of such are complex and real cobordisms which have been used to classify the stronglyinteracting invertible fermionic topological phases [26].
In this paper, we utilize the mathematical notion of equivariant "spectra" and the theory of equivariant stable homotopy, which further generalize the notion of a cohomology theory and unveils deeper relations within the symmetry classes of quantum systems; see Refs. 27  Table I. and 28. Similar notions had been surfacing in various areas of physics over the past decade, notable examples include: Kitaev [29][30][31], Freed and Hopkins [26,32], Kapustin et al. [33,34] and Chen, Gu, Liu, and Wen [35,36]; for an overview see Xiong at Ref. 37. Moreover, as the modern mathematical framework for algebra in spectra has also been building up over the past decade (a comprehensive treatment is provided in Ref. 38), we believe this modern formulation may shed light on various fur- ther new and exciting topics across physics. We move on to show how to use equivariant spectra in order to perform a unified analysis of crystalline TISC (CTISC) in all ten AZ symmetry classes, and obtain a complete classification of topological phases as well as anomalous surface states in various key crystalline space-group symmetries; see Table II in Sec. III. Over the past several years, many naturally inquired the effects of spatial crystalline symmetries of CTISC on their topological classification . A major advancement in that topic came with the formulation of symmetry indicators (SI) and topological quantum chemistry which enable the detection of topological phases from the crystalline symmetry properties of the band structure [25,[64][65][66][67][68][69][70][71][72][73][74][75][76][77]. These have contributed to the discovery and understanding of diverse topological phenomena such as higher-order TISC (HOTISC) [74,[78][79][80][81][82][83][84][85][86][87][88][89][90][91][92], "fragile" TISC [66,[93][94][95], obstructed atomic limits [66,95], and boundary-obstructed TISC [96].
The majority of classification efforts have naturally focused on time-reversal invariant topological insulators (TIs) with spin-orbit coupling, i.e., AZ class AII; see Fig. 1. Nevertheless, our understanding of topological crystalline phenomena has since expanded to incorporate other AZ classes. However, it has been known that some CTISC are not detectable by their SI and that other topological invariants (such as Berry phases) are needed to discern them, this has been dubbed surface state ambiguity. Particularly, for two superconducting AZ classes, AIII, DIII, and CI, no gapped topology is detectable by SI; see discussion in Sec. III B 5. Noticeably, over the past couple of years, two independent works by Khalaf et al. [25] and Song et al. [63] have presented a classification of anomalous surface states for AZ class AII using different methods.
In this paper, we show that the topological phases of CTISC are all manifestations of an underlying spectrum for each space group symmetry. We first derive these spectra and harness their properties to gain a unified description of the K-theory and the topological invariants amongst all ten AZ symmetry classes. We further use spectra to gain an understanding of atomic insulators (AI) and thus of anomalous surface states which may be regarded as their complements.
Before reading on, one should note that equivariant spectra are not to be confused with the related notion of "spectral sequences", which have also been used to describe CTISC phenomena [92,[97][98][99].
The rest of the paper is organized as follows: In Sec. II, we overview the classification problem of CTISC and present the essence of the equivariant spectra paradigm. In Sec. III, we present our main classification results and outline the main methods used to derive them. In Sec. IV, we take a "hands-on" approach and provide detailed derivation via a thorough study of a pedagogical example. We conclude in Sec. V.

A. Crystalline topological insulators and superconductors
The classification of all quantum states protected by symmetries of the system is one of the main challenges facing the condensed matter physics community. Even in the absence of any crystalline symmetries, there is a plethora of topological phenomena including many exotic states of matter [100]. In this work, we focus our efforts on the classification of topological states of weaklyinteracting fermions. Their symmetries in general are split into internal non-spatial symmetries and spatial symmetries such as crystalline symmetries. Before moving on, we note that in the absence of any crystalline symmetries, major progress was recently made in the classification of "invertible" topological phases of stronglyinteracting fermions [26]; in Sec. V B 1, we discuss how this may be expanded using the formalism described in this paper.
In presence of crystalline symmetries, there is already a vast phenomenology of topological phases including HOTISC, fragile TISC, obstructed atomic limits, and boundary-obstructed TISC. One of the remarkable properties of the TISC in absence of crystalline symmetries is that they all fit within the same systematic periodic table of TISC; see Table I. These diverse phases include: integer quantum Hall (IQH) phases [101,102]; time-reversal symmetric quantum spin-Hall phases and strong/weak TIs [5, 15-18, 20, 22, 103]; topological Majorana bound states and p x + ip y topological superconductors (TSCs) [104,105]; and many others. It is thus extremely desirable to treat CTISC phenomena in a similar unified manner; this is what we are set to do.
One says two quantum states belong to the same topological phase if they can be adiabatically deformed into one another while respecting the symmetries of the system and without encountering a phase transition (i.e., equivariantly homotopic). We focus on either insulators or superconductors where the groundstate is character- ized by the existence of an energy gap. The many-body groundstate of a free fermion gapped system is given by the choice of filled valance bands and empty conduction bands. There is a natural notion of band addition by stacking bands together. Moreover, in the study of band representations, one often also considers band subtraction. This is known as the "stable" limit, within it, as we explain in Sec. II B, the classification of all topologically distinct groundstates is given by (twisted) equivariant Ktheory [89,90,92,97,[106][107][108][109][110][111][112].
In this paper, we provide forceful techniques that allow us to carry out explicit calculations of the full K-theoretic classification. However, easily attainable partial knowledge of the topology of a state is often given in terms of its SI. We believe it is useful to first explain the relations between K-theory and SI.
At each point in the Brillouin Zone (BZ), k ∈ BZ, one can always decompose each band according to representations of the symmetry group preserving that point. This can be viewed as a quantum state of an effective 0dimensional (0D) system at k. The topological invariants of each representation at each point are called "band labels". However, the fact that these band labels originate from a state on some d-dimensional BZ naturally imposes certain compatibility conditions; for example, continuity requires the labels to remain constant along highsymmetry lines within the BZ. It is thus natural to study physical band structures, BS TISC , as band labels modulo physical compatibility relations. In order to avoid possible confusion, we note that there exists variability in the literature regarding which compatibility relations to impose in the definition of BS, this may lead to an inclusion of (semi-)metallic states within BS (see, e.g., discussion in Refs. 65, 72, and 75); we focus our attention on TISC and hence use the stricter definition such that BS TISC only includes band labels corresponding to gapped systems, i.e., insulators and superconductors [113].
Each distinct class of topologically equivalent TISC groundstates is represented by a distinct member of Ktheory. Since BS TISC are topological invariants, the band labels of every member of such an equivalence class would be identical. Therefore, two quantum states with different band labels are necessarily topologically distinct, and there exists an onto map (epimorphism), K → BS TISC . However, the converse is in general not true; there often exist two quantum states which have identical band labels but are nevertheless topologically distinct [25,69]; see Fig. 2.
A compelling feature of TISC is bulk-boundary correspondence and the existence of anomalous surface states. This persists in CTISC; the boundary of a gapped insulator or superconductor which is not an atomic insulator or superconductor (AI) hosts either strong, weak, or higher-order anomalous surface states. We emphasize that in order to keep with the conventional jargon we use AI to denote both atomic insulators and atomic superconductors. Therefore, in order to classify anomalous surface states one broadens the topological equivalence such that two quantum states are considered equivalent if they only differ by stacking AIs. The topological invari-ants of AIs within K-theory, AI → K, are thus quotiented out and the anomalous surface states, ASS, are given by ASS = K/AI. By construction, this also nullifies fragile CTISC and obstructed atomic limits, the study of which can be done by solely inspecting the structure of AI [66,93,95].
A similar construction is standardly carried for BS TISC to obtain the symmetry indicators of TISC, SI TISC = BS TISC /AI. As discussed above, generically, two quantum states which have identical SI may still have topologically distinct anomalous surface states [25,69]. Therefore, SI serve as easy-to-compute useful topological invariants which approximate the full K-theoretic topological classification of anomalous surface states. The above discussion is neatly summarized by the following commutative diagram: This is depicted in Fig. 2. As a means of better approximating the K-theory, one may add non-local invariants such as Berry phases to the SI [66,69]. However, to the knowledge of the authors, no proof is yet given that the currently known invariants capture all possible CTISC phases in all AZ symmetry classes encapsulated by Ktheory.
In the following sections, we briefly provide the essential basics of the full K-theoretic topological classification and how spectra may be utilized to resolve it.

B. Topological classification
For simplicity, we initiate our discussion of topological classification by considering systems with no symmetry (other than charge conservation). As discussed above, the many-body groundstate of a free fermion gapped system, |ψ(k) , is given by the choice of filled valance bands and empty conduction bands. For a system with n bands, one has to first specify the number of filled bands, m = 0, . . . , n. Next, one has to pick which bands are filled and which are empty; this is equivalent to specifying a basis, i.e., a U(n) matrix. Recall that two quantum states belong to the same topological phase if they can be adiabatically deformed into one another without closing the energy gap, therefore, the exact basis choices within either the valance or the conduction bands, i.e., U(m) and U(n − m), are redundant. Thus, the space of topologically distinct basis choices is denoted C 0 (n) = m U(n) U(m)×U(n−m) . The groundstate of a translationally invariant quantum system is given by continuously making these basis choices for every point in the BZ, k ∈ BZ, i.e., picking a continuous map |ψ(k) ∈ Map(BZ, C 0 (n)), where Map(X, Y ) is the space of maps from X to Y . Therefore, the set of topologically distinct groundstates is given by, π 0 (Map(BZ, C 0 (n))). Here, π 0 (Y ) is the set of topologically connected components of a space Y . Intuitively, the addition of trivial filled or empty bands should not drastically affect the groundstate of a quantum system. This intuition manifests itself in the mathematical notion of stability. Particularly, for any n, n large enough, one finds Map(X, C 0 (n)) Map(X, C 0 (n )); this is noted as the "stable limit" [114], Topologically distinct stable groundstates = π 0 (Map(BZ, C 0 )), where C 0 def = lim n→∞ C 0 (n) is denoted the "classifying space"; see Table I.
Although in general, the set of topologically distinct groundstates is not required to have any algebraic structure, the stable limit always yields a set with an abelian group structure. This observation is deeply related to an alternative route leading to the same classification. As discussed in Sec. II A when studying topological BS one often allows formal differences of quantum states. This in-fact enforces an abelian (additive) group structure on the space of states, which is the defining property of Ktheory, Indeed, as explained in further detail in Appendix B 2, one has KU 0 (X) = π 0 (Map(X, C 0 )) for any topological space X. Henceforth, we focus our attention on stable groundstates. In the following sections, we consider the effects of symmetries on this classification scheme, where the Ktheoretic classification becomes hard to compute. This is also where the relations between K-theory and equivariant spectra become more apparent.

Symmetry & equivariance
In the presence of symmetries, quantum states are only considered equivalent if they can be adiabatically deformed into one another without breaking the symmetries. One thus has to refine the analysis of the previous section. The group of topologically distinct groundstates is given by equivalence classes of maps from the BZ to the classifying space which respect the symmetries; such maps are called "equivariant" maps.
The ten AZ symmetry classes correspond to the presence or absence of the non-spatial symmetries. These are the two antiunitary symmetries, time-reversal symmetry, Θ, particle-hole anti-symmetry, Ξ, as well as the unitary sublattice/chiral anti-symmetry, Π; see Fig. 1. In addition to these, a crystalline space-group symmetry acts via its point-group, G, on the BZ. In this paper we focus on non-magnetic space-groups where the crystalline symmetries act independently of the non-spatial symmetries; see Sec. V B 2 for further discussion.
The simplest symmetry is sublattice/chiral symmetry, Π, which exchanges the filled and empty bands and thus acts only on the classifying space. This restricts the classifying space to U(n) = C 1 (n) ⊂ C 0 (2n) which is fixed under the symmetry. The group of topologically distinct groundstates in the absence or presence of Π is therefore given by equivariant K-theory which corresponds to equivariant maps, Here, as in Table I, q = 0, 1 correspond to AZ classes A, AIII, respectively; these are denoted the two complex AZ symmetry classes. The situation becomes rather more intricate when considering the presence of antiunitary symmetries. We thus reserve the detailed discussion to Appendix B 2; for a nice overview see Ref. 115. Both Θ and Ξ represent an antiunitary, Z T 2 -symmetry and in-fact differ only by their action on C q . There are eight possible antiunitary actions, each of which corresponds to choosing a different real structure on C q . These are denoted the eight real AZ symmetry classes and they are classified by real equivariant K-theory, Here, as in Table I, q = 0, . . . , 7 correspond to AZ classes AI, BDI, D, DIII, AII, CII, C, CI, respectively. Note that C q and C q mod 2 are isomorphic as topological spaces, but differ by the action of Z T 2 . This becomes more apparent in the following example: Consider a space, X, which is invariant under the antiunitary Z T 2 -symmetry. This can be, for example, either some parameter space of the system or a space surrounding a defect [8,11]. Similar to the sublattice/chiral case, the Z T 2 -symmetry now acts to restrict C q to its fixedpoints space under the symmetry, R q ⊂ C q , such that The fixed-points spaces, R q , are denoted the real classifying spaces and explicit expressions for them are given in Table I. Before moving on, we note that the notation "Z T 2 " for the antiunitary Z T 2 -symmetry originates from its manifestation in classes AII and AI as time-reversal symmetry. Therefore, even though we provide a unified classification for all ten AZ classes, which includes manifestations as particle-hole symmetry, we stick to this notation in order to comply with the accepted dogmas.

Spectra -an introduction
In general, equivariant K-theory [Eq. (5)] is not easy to compute. In this paper, we utilize the notion of equivariant spectra and use it to obtain explicit results. In order to acquaint the reader with this powerful notion, we provide some properties of K-theory and spectra which provide some useful intuitions.
Consider a 1-sphere (circle), X = S 1 , in AZ class, q, which is invariant under Z T 2 . The topological classification, KR −q,0 (S 1 ), is given by maps from S 1 to R q . When examining the space of maps, one notices that R q are not always path-connected (e.g., R 1 (n) = O(n) has two distinct connectivity components discerned by the sign of the determinant). Therefore, one must first pick a basepoint within one of the connected components and then a map from the 1-sphere starting in that base-point. The latter is the definition of the fundamental group, π 1 , and hence The based loop-space of Y , denoted ΩY , is the space of all based loops within Y , such that Remarkably, all the classifying spaces C q and R q are all infinite loop-spaces such that Here, q ∈ Z and one has R q R q mod 8 . This unique property is a manifestation of Bott-periodicity [116,117] and is at the heart of the periodic table of TISC. This also brings us to spectra.
The notion of spectra has considerably evolved over the past decades, see Refs. 38 and 118. Classically, a spectrum is a sequence of spaces, X i , which satisfy X i = ΩX i+1 . We have thus already encountered two examples of such spectra: First, the KU spectrum defined by the (2-periodic) sequence X i = C −i . Second, the so-called KO spectrum defined by the (8-periodic) An equivariant spectrum is similarly classically defined as a sequence of spaces, X i = ΩX i+1 with an action of a group G; these have to satisfy some compatibility relations discussed in Appendix B 3. One of the main equivariant spectra of interest in this work is the KR spectrum. It is defined by the 8-periodic sequence X i = C −i with the action of Z T 2 such that the fixed-point spaces are given by There is a natural internal group operation for loopspaces given by concatenating two loops, where the inverse is given by reversing the orientation. Furthermore, since a spectrum consists of infinite loop-spaces one can show that this group operation must be commutative up to a continuous deformation, i.e., a homotopy; see Fig. 3. This implies that every spectrum is endowed with an abelian additive group structure. Specifically, any two points x, x ∈ X i may be added or subtracted such that x ± x ∈ X i is well-defined and unique up to homotopy.
As a consequence, many properties of abelian groups are also respected by spectra, this often poses a major simplification. In particular, the space of G-equivariant FIG. 3. Two concatenated loops on any topological space with a group structure always commute up to a homotopy. Here, such a homotopy is depicted for two loops (red and blue) on an abstract torus, T 2 U(1)×U(1); three intermediate stages of this homotopy are presented. maps between two G × Z T 2 -equivariant spectra E, E can be thought of as the "abelian group" of G-equivariant homomorphisms Hom G (E , E) which is by itself a Z T 2equivariant spectrum. Moreover, for any equivariant space, X, one can construct a spectrum S[X] such that [119] Map This implies that the spaces of all quantum groundstates of a crystalline system for all real AZ symmetry classes can be expressed by one Z T 2 -equivariant spectrum, The complex AZ classes are similarly captured by an equivariant spectrum denoted KC G . Furthermore, the atomic insulators also form a spectrum and therefore the map AI → K whose cokernel yields the topological phases with ASS [see Eq. (1)] also stems from a homomorphism of spectra. This mindset of describing everything in terms of equivariant spectra allows us to perform algebraic abelian calculations that capture all AZ symmetry classes at once and yield explicit results we hereby present.

III. MAIN RESULTS & METHODS
The formulation of topological physics in terms of equivariant spectra provides a useful tool for exploring hidden relations of topological states of quantum matter. We hereby provide explicit quantitative results that demonstrate these qualitative ideas.
Concretely, we first use the properties of the spectra formalism to obtain the complete classification of topologically distinct quantum states of 3D CTISC in all AZ symmetry classes for key space-groups. Second, we utilize other aspects of spectra to obtain an understating of the nature of AIs in the K-theory classification and provide a complete classification of topological phases with anomalous surface states for these space groups; see pictorial depiction in Fig. 2(left).
The classification groups of topological phases are summarized in Table II; detailed results including the full Ktheory as well as 2-dimensional (2D) layer-groups and 1-dimensional (1D) rod groups are presented in Appendix A.
The complete mathematical derivation will be provided in Appendix B. Nevertheless, we hereby outline the essence of the methods used to obtain these results in Secs. III A and III B.

A. Topological classification and distinct states
Similar to the discussion in Sec. II B 2, when classifying all equivariant maps from the BZ to the classifying spaces one must first specify a connected component of the classifying space and then focus on maps within that component. This entices the definition of the "reduced" K-theory, KR, where "pt" is the space comprised of a single point; cf. Eq. (7). This decomposition has a physical interpretation: Consider an AI originating from the trivial Wyckoff position [x = (0, 0, 0)]. The groundstate corresponding to this AI has identical band structure at all points (momenta) in the BZ and hence is fully determined by KR G (pt). The treatment of AIs at other Wyckoff positions is rather more complicated and is elucidated at Sec. III B. However, in the absence of crystalline symmetries the only Wyckoff position is the generic Wyckoff position, which therein gives the same contribution as the trivial position. Hence, it is very easy to distinguish between AIs which are classified by KR(pt), and topological phases with anomalous surface states which are classified by KR(BZ). For each AZ class, the AIs satisfy KR −q,0 (pt) = π 0 (R q ) and are easily read from the d = 0 column in Table I. In general, determining the topological phases encapsulated by KR G (BZ) is a nontrivial task, whose solution we obtain using equivariant spectra. In order to acquaint the reader with this formalism, we first apply it to re-derive the classification of topological phases in the trivial space-group, P1, see the first record of Table II. Historically, this was achieved by various methods such as the Baum-Connes isomorphism and the Poincaré duality [2]. However, as we explain below the spectra perspective is generalizable to non-trivial space-groups.

First steps with spectra
Geometrically, the BZ of a 3D system is a 3-torus, BZ =T 3 =S 1 ×S 1 ×S 1 , with an involutive action, i.e., TABLE II. Complete classification of topological phases with anomalous surface states of 3-dimensional crystalline topological insulators and superconductors for key point-groups (PG) and space-groups (SG) in all ten Altland-Zirnbauer symmetry classes. Each entry corresponds to the total group of all strong, weak, and higher-order surface states; more detailed results are provided in Appendix A. Point-groups are given in Schönflies notation. Space-groups are given in Hermann-Mauguin notation and specified by their number as given in Ref. 120.
4. An equivariant decomposition of the Brillouin zone torus into eight cells centered at high-symmetry momenta; see Eq. (14).
k → −k, of the Z T 2 -symmetry on the momenta, k ∈ BZ. Instead of studying the BZ torus itself, we examine its "free spectrum", and S is hence referred to as the "sphere spectrum". This decomposition provides the first hint for the usefulness of spectra, as the K-theory base-point decomposition [Eq. (12)] is now evident already at the geometric level. Note, that the notation ΣE ≈ Ω −1 E in spectra theory is analogous to (but is not to be confused with) the classical suspension, ΣX, of homotopy theory [121]. The sphere spectrum satisfiesΣ d1 S ⊗Σ d2 S Σ d1+d2 S; thus, gathering all the above, we obtain There is a very intuitive interpretation of this decomposition associated with some high-symmetry momenta (HSM): In Fig. 4 we see a decomposition of the 3-torus BZ as a Z T 2 -CW complex. In general a G-CW complex is a space composed of disjoint d-dimensional open cells such that each cell is mapped by the symmetry, G, to a cell of the same dimensionality. Here, we label the cells composing the BZ by the HSM at their centers. The 3-torus decomposes into eight cells, one 0D cell at the Γ point, three 1D cells centered at the X, Y, Z points, three 2D cells centered at the S, T, U points, and one 3D cell centered at the R point [122]. Each of these ddimensional cells is associated with aΣ d S spectrum in Eq. (14).
The distributivity of homomorphisms, (15) together with the spectra to K-theory relation, π q−p Hom(Σ d S, KR) KR p,q (S d ), immediately turns Eq. (14) to a K-theory decomposition, Here, the K-theory of the spheres,S d , are much simpler objects and follow directly from Bott-periodicity [2,116,117], (17) These are easily read from Table I. For example, in AZ symmetry class AII, we have q = 4 and thus find, cf. P1 in Table II. Here, the Z atomic insulators correspond to the number of Kramer's pairs, while the Z 3 2 correspond to the weak TIs and the last Z 2 to the strong TI. The strong TISC are phases reflecting the topology of the bulk of a material and are indeed originate from the R-point cell; see Fig. 4. The weak TISC phases, which are here associated with the S, T , and U -point cells, can be constructed by stacking 2D TISC along these directions. We reemphasize that, in the absence of crystalline symmetries, there are more elementary ways to calculate the topological phases of the BZ torus. However, as we immediately show, the spectra perspective generalizes very well to the crystalline case.

Crystalline symmetries & equivariant spectra
The key aspect revealing the structure of topological phases is that the free spectrum of the BZ torus itself decomposes into sphere spectra. This decomposition then implies that the classes of topologically distinct groundstates, represented by the K-theory of the BZ torus, decompose to classes within the K-theory of spheres which are much better understood. The equivariant spectra perspective reveals that this reasoning applies to crystalline systems as well.
In a previous work by one of the authors with A. Chapman, see Ref. 62, an explicit classification of Dirac Hamiltonians invariant under any point-group symmetry, G, was provided. In our current settings, these classes of Dirac Hamiltonians correspond to topologically distinct groundstate classes associated with a G-equivariant sphere. In K-theoretic language, the classification by E. C. and A. Chapman is provided by an isomorphism between the K-theory of G-equivariant spheres and the non-equivariant K-theories of spheres corresponding to different representations of the point group G. Explicitly, (20) Here, Irr Z2 (G) are all Z 2 -graded and -twisted irreducible representations for suitable grading and twisting explicitly determined by the point-group, G; the degrees, p ρ , q ρ , are also explicitly determined. A proof of this isomorphism was given by M. Karoubi in Ref. 123.
A question arises, whether the topologically distinct groundstates of crystalline systems, represented by KR G (T 3 ), can also be decomposed into equivariant spheres and thus, via the above isomorphism, to noncrystalline invariants.
We prove that this is indeed true for a class of spacegroups presented in Table II. The proof is given in Appendix B 4 and becomes very straightforward once formulated in terms of equivariant spectra. Let us present the intuition behind this decomposition.
One of the key aspects of the non-crystalline decomposition in Sec. III A 1 was the decomposition of the BZ torus to spheres,T 3 =S 1 ×S 1 ×S 1 , where each sphere corresponds to one of the primitive reciprocal lattice vectors, b 1 , b 2 , b 3 . However, a G action may in general act on the BZ and map a primitive sphere into a nonprimitive loop within the BZ. This corresponds to an equivalent but different choice of primitive vectors. Let us consider the class of space-groups where there exists a choice of primitive vectors such that the point-group, G, respects this choice, i.e., for any g ∈ G and any b i there exists b j such that g(b i ) = ±b j . This means that the point-group acts as a signed permutation on the primitive lattice vectors, and thus preserves the direct product structure of the BZ torus; we dub this class of spacegroups "signed-permutation representations".
As proven in Appendix B, we find that the free G × Z T 2 -equivariant spectrum,T 3 , of the BZ torus of a signed-permutation representation decomposes into G kequivariant sphere spectra, where, the "little groups", G k ⊆ G stabilize the HSM, k ∈ BZ, located at the centers of d k -dimensional cells (points, lines, planes, and volumes) spanned by the primitive reciprocal lattice vectors [124] For example, in a primitive cubic lattice, {k} are the Γ, X, M, R-points, while in a base-centered orthorhombic lattice, {k} are the Γ, S, Z, R, Y, T -points [122]. This is analogous to the simpler Z T 2 -equivariant case discussed above; cf. Eq. (14) and Fig. 4. The induction from G k to G in Eq. (21), which is a standard procedure in the study of band representations, here occurs already as a description of the BZ geometry.
As before, by using the spectra formulation, this decomposition which happens purely at the geometric level, immediately translates to the K-theory classification,  (20). We thus obtain our first quantitative result -an explicit formula for the full classification of all topologically distinct groundstates of all signed-permutation CTISC in all AZ symmetry classes; the classification tables are presented in Appendix A. The AIs at the trivial Wyckoff position are still captured by the Γ-point summand, KR GΓ (S dΓ ) = KR G (S 0 ) KR G (pt). However, the AIs at other Wyckoff positions are not so easily isolated. This is discussed in the following section.

B. Topological phases and surface states
In order to obtain the full classification of topological phases with anomalous surface states one must quotient out the AIs from the full K-theory classification of topologically distinct groundstates obtained above; see pictorial depiction in Fig. 2(left).

Atomic insulators & the ai map
An atomic insulator state, by definition, may always be adiabatically connected to a state with localized Wannier orbitals around specific Wyckoff positions. Each Wyckoff position, x, is invariant under a symmetry with point group G x ⊆ G. As such, the Wannier orbitals of the AIs at x must transform as representations, ρ, of G x . In general, two AIs that differ either by their Wyckoff positions or by their representations of G x cannot be continuously deformed into each other. They may hence correspond to different classes of topologically distinct groundstates and thus different elements of the K-theory. Therefore, the AI groundstates must be quotiented out in order to obtain the topological phases with anomalous surface states.
An AI formed by a Wannier orbital with a particular representation at a particular Wyckoff position is often referred to as an elementary band representation. Such elementary band representations form a generating set for the AI additive group. This provides a concise description at AZ symmetry classes A, AI, and AII which form Dyson's threefold-way [14,125], i.e.,

AI AZ classes
abelian groups x ρ∈Irr(Gx) Z. (23) Here, Irr(G) are the irreducible representations of G which form the basis of the representation ring, Rep(G), consisting of all representations of G with multiplication given by the tensor product of representations. This multiplicative structure would prove vital ahead. Nevertheless, the generating set of irreducible representations is rather redundant for the description of the other AZ symmetry classes which unveil a deeper algebraic structure. The constraints imposed by the presence of either particle-hole or chiral symmetry (see Fig. 1) imply that stacking identical AIs may be topologically equivalent to filling no Wannier orbitals. An AI at a particular Wyckoff position, x, is a 0-dimensional system and is hence classified by the K-theory of a point, KR Gx (pt). This automatically quotients-out the topological redundancies.
For example, the AIs at the generic Wyckoff position in space-group P1 are given by KR −q,0 (pt) at AZ symmetry class, q. The particle-hole and chiral symmetries reduce the Z AIs of Dyson's threefold-way (AZ classes A, AI, AII) to either Z 2 or 0 for the other AZ classes; see Table I. This has been recently utilized in Refs. 75-77 for the study of SI of superconducting systems.
The AIs in any space-group symmetry decompose according to the Wyckoff positions, For any AZ class, q, these are explicitly given by [126] KR −q,0 KR −q−4,0 (pt). (25) This generalizes Eq. (23) to all AZ classes and indeed, In order to gain an intuition for the shift of 4 for quaternionic representations in Eq. (25), recall that a quaternionic representation may be expressed in terms of unit quaternions which form an SU(2) group. The four AZ classes, q = 2, 3, 4, 5 (D, DIII, AII, and C), all have broken spin rotation SU(2) symmetry, while the other four AZ classes (C, CI, AI, and BDI) are all invariant; a quaternionic representation thus exchanges these two sets of classes. See Ref. 115 for further detail.
Before we can quotient out the AIs in order to find the topological phases with anomalous surface states, we must first find the map which evaluates the AIs as equivalence classes in the K-theory classification of topologically distinct groundstates, cf. Eq. (1). We denote this map by ai.
On the most basic level, we may treat this map as an abelian group-homomorphism, i.e., evaluating the AIs at a particular AZ symmetry class and returning the group element of their topological classification. However, much more structure is revealed when treating ai as a homomorphism on the level of module-spectra. We find that the contribution of any Wyckoff position to all AZ classes at once is captured by (at most) two integers per Z 2 -graded representation.

Immediate implications for anomalous surface state
Before elaborating on the ai map, we note an immediate consequence of Eq. (25). By explicitly plugging the abelian groups from the d = 0 column in Table I, one sees that AI = 0 for AZ classes AIII, DIII, and CI for all Wyckoff positions of any space-group symmetry. This implies that K = K/AI = ASS and hence: In AZ symmetry classes AIII, DIII, and CI, non-trivial topologically distinct groundstates of any weakly interacting fermionic crystalline system in any spatial dimension, all have anomalous surface states and correspond to nontrivial topological phases.

Shiozaki's formula
Consider the contribution of a single HSM summand in Eq. (22), classified by KR G (S d ). In a recent work, K. Shiozki showed that AIs at the center of the pointgroup, G, can be realized as Dirac Hamiltonians with a spatially dependent "hedgehog" mass-term; see Ref. 127. In K-theory language, this is captured by the identity KR G (pt) KR G (S d ∧ S d ), which is an equivariant version of Bott-periodicity. Here,S d ∧ S d is a G × Z T 2equivariant 2d-dimensional sphere, such that the former d coordinates describe the Z T 2 -odd momenta around the Dirac point and the latter d coordinates describe the Z T 2even real-space dependence. The topological class of the groundstate of such a Dirac Hamiltonian may be readily found by neglecting the "hedgehog" dependence of the mass-term. This understanding enabled K. Shiozaki to obtain an explicit formula for the ai map. In K-theory language, this stems from the (equivariant) embedding of the d-sphere within the 2d-sphere, The isomorphism of Eq. (20) provides an explicit basis for this equivariant K-theory map in terms of (Z 2 -graded) representation theory. Results for all magnetic and nonmagnetic 3D point-groups are provided in Ref. 127. For the signed-permutation representations, discussed in Sec. III A 2, we find that the AIs contribution to a HSM component, KR G k (S d k ) ⊂ KR G (BZ) may be distilled (see Appendix B 6) to the AIs at the Wyckoff position, x, reciprocal to k. For example, in a primitive cubic lattice, the reciprocal to the R-point, k = (π, π, π), is the Wyckoff position, x = ( 1 2 , 1 2 , 1 2 ). These are the AIs at the center of the point group, and one has G x G k [128]. This implies that Shiozaki's formula may be utilized to compute the full ai map using our spectra decomposition, Eq. (22).
Let us thus focus on a particular HSM with a pointgroup, G, for which the ai map reduces to Eq. (28).

Ring-spectra & multiplication of quantum states
A hint of a multiplicative structure already appeared with the identification KR 0,0 G (pt) Rep(G) [Eq. (26)] with multiplication given by the tensor product of representations, e.g., Irr( where E is the complex representation and 1 is the trivial representation. It is often overlooked that such a structure can be extended to a multiplication of quantum states. Any two quantum groundstates of a system may be thought of as vector bundles over the BZ; this endows them with a multiplication given by the graded tensor product of the bundles. Particularly, since KR G (pt) ⊂ KR G (X) for any space, X, this implies that elements of all equivariant K-theories considered so far may be multiplied by KR G (pt), i.e., Crucially, this product structure mixes different AZ classes in an additive manner. Nonetheless, for X = pt and p, p , q, q = 0 it reduces to the multiplicative structure of Rep(G) discussed above. Note, that a similar module-structure for KU was discussed by Shiozaki, Sato, and Gomi in Ref. 111. The multiplication on K-theory stems from an inherent multiplicative structure on the spectra KR and KR G , which, by definition, make these into ring-spectra. The ai map may thus be considered as a modulehomomorphism, i.e., ai(u · v) = u · ai(v). This has two important consequences that reveal the structure of the ai map.
First, it provides us with a simple expression for the most general structure of the map from an AI of a real irreducible representation u ∈ KR(pt) ⊂ AI G to each real irreducible graded representation, ρ, in Eq. (20), Hence, for all AZ classes, the single element v ρ ∈ KR pρ,qρ (S d ) which is independent of the AZ class, determines the ai map. This element is itself determined by (at most) one integer corresponding to the classification of d spatial dimensions at level q ρ − p ρ in Table I. Mathematically, this immediately follows from the fundamental identity, Hom KR (KR, KR) = KR. The analyses of complex and quaternionic representations are similar yet slightly more complicated. In particular, since any representation has a complex conjugate representation, this at most doubles the number of integers; see Appendix B 3 e. Second, the module-homomorphism structure of the ai map implies that the topological classification of all AIs in any representation, u ∈ AI G , is determined by the classification, ai(1), of the fundamental AI corresponding to the trivial representation, We thus conclude that the multiplicative structure enables us to determine the topological classification of any AI of any orbital by the fundamental AI of an s-orbital at the same Wyckoff position. The latter is readily computed using Shiozaki's formula. This is how we obtain all the results presented in Table II and Appendix A.

Immediate implications for symmetry indicators
As a final remark, the same reasoning as in Sec. III B 2, that enabled us to exclude the existence of AIs at AZ classes AIII, DIII, and CI, applies for SI as well. The band labels correspond to individual HSM points, k ∈ BZ, and thus in gapped systems, they are also classified by the K-theory of a point, where the compatibility relations relate the invariants along high-symmetry lines, planes, and volumes in the BZ. Regardless of these compatibility relations, one sees that BS TISC = 0 for AZ classes AIII, DIII, and CI for all Wyckoff positions of any space-group symmetry. This implies that SI TISC = BS TISC /AI = 0 and hence: In AZ symmetry classes AIII, DIII, and CI, all non-trivial topological phases with anomalous surface states of any weakly interacting fermionic crystalline system in any spatial dimension cannot be detected by symmetry indicators; see Table II. Any non-trivial SI must indicate a gapless state and not a CTISC.
Our methods were implemented as a GAP4 [129] language algorithm. However, we believe it is highly beneficial for the reader to see some of the finer details. First, in Sec. IV A we provide the basics of the hidden multiplicative structure within the periodic table of TISC. Then, in Sec. IV B, we study a "hands-on" pedagogical example, which portrays the use of our methods from top to bottom, i.e., choosing a particular space-group and fully deriving its complete classification of topologically distinct groundstates and topological phases with anomalous surface states. Finally, in Sec. IV C, we use this example and demonstrate how to construct model Hamiltonians for the numerous topological phases in Table II.

A. Multiplication tables of TISC
As discussed in Sec. III B 4, the multiplicative structure of the K-theory and its inherent ring-spectrum, is at the core of our understanding of anomalous surface states. The first hint of multiplicativity was already provided by KR 0,0 G (pt) Rep(G). However, before one is ready to fully tackle a computational example, a more explicit sense of the ring-structure would be advantageous.
Therefore, we hereby discuss the (multiplicative) rings, These rings and the maps between them, convey hidden aspects of the inherent ring-spectra and of the periodic table of TISC. Using, KR −q,0 (S d ) = KR −q,−d (pt), as discussed in Sec. III A 1, these rings consist of all the different Z and Z 2 invariants of the periodic table of TISC in Table I.  Table III.
In order to discern these different Z and Z 2 invariants, we label their generators by 1, η, η 2 , α, as summarized in Table III. The twofold and eightfold Bott-periodicities are captured by the invertible elements, ξ and β, respectively; the diagonal Bott-periodicity is captured by the invertible element, µ. For example, the strong time-reversal invariant TI of real AZ class AII with q = 4 and d = 3 may be located in Table I by starting at the Z 2 · η invariant of q = 1 and d = 0, and moving 3 diagonal steps down; it is thus represented by ηµ 3 . Similarly, the IQH effect of complex AZ class A with q = 0 and d = 2 may be located in Table I by starting at the complex Z invariant of q = d = 0, moving 2 diagonal steps down and applying the twofold Bott periodicity up; it is thus represented by ξ −1 µ 2 .
The multiplication table of KR * , * (pt) is presented in Table IV. This multiplication table has some familiar features, such as 2η = 0 and 2η 2 = 0 which encapsulate the Z 2 nature of the corresponding topological phases, such as strong and weak time-reversal invariant TIs and Majorana bound states. Moreover, this multiplicative structure also has some intriguing consequences on CTISC.
The simplest example for such a direct consequence is the centrosymmetric space-group, P1, where the pointgroup, G Z 2 , acts by inversion, i.e., k → −k. The strong CTISC component of its ai map is given by where, Particularly, for AZ class AII we have q = 4 and hence, where we have set v = α and used β −1 α · α = 4, see  The primitive unit-cell of space-group P4. The Wyckoff positions, a, b, c, d, g, reciprocal to the high-symmetry momenta are marked. Note that the g Wyckoff position is reciprocal to both the X-point and the R-point HSM.
space-group P1, which may be re-interpreted as a direct manifestation of the multiplication table, Table IV.
We reserve the analyses of other examples to the following sections and Appendix B. Nevertheless, we would like to emphasize a vital property of the multiplicative structure which stems from the inherent spectra: There are natural maps of realification, r, and complexification, c, between the spectra KR and KC. Crucially, since the ai map is not just a K-theory map, but rather a map between the inherent spectra, any ai map between any real or complex K-theories must be given by compositions of r, c, and multiplication by elements of the K-theories. The manifestations of the realification and complexification maps on the K-theories is provided in Table III. In particular, r(1) = 2 stems from C R ⊕ iR and c(1) = 1 stems from C ⊗ R C. For a complete list of possible maps see Table XVI in Appendix B. B. A pedagogical example: space-group P4

Geometry
Consider a crystalline material of space-group P4 with tetragonal-disphenoidal S 4 point-group symmetry. The primitive unit-cell may be chosen as a parallelepiped with edges along its primitive lattice vectors. Here, this forms a square cuboid depicted in Fig. 5 (bottom). Similarly, the BZ may also be chosen as a parallelepiped with edges along its primitive reciprocal lattice vectors, b 1 , b 2 , b 3 . This forms the square cuboid depicted in Fig. 5 (top).
The tetragonal-disphenoidal S 4 point-group symmetry is an abstract fourfold G Z 4 symmetry which acts on the primitive reciprocal lattice vectors by with a similar action on the primitive lattice vectors. However, since we study spinful-electrons, one must take into account the (projective) spin-representation (ŝ 4 ) 4 = −1. The easiest way to deal with this inconvenience is to considerŝ 4 as a generator of the double point-group G Z 8 eightfold symmetry with (ŝ 4 ) 8 = 1. We shall use this description from here on. The building blocks of the topological classification correspond to the six HSM, k, with π-integer coordinates, Γ, Z, X, R, M, A. Each of them is stabilized by its little groupĜ k , these are listed in Table V and depicted in Fig. 5 (top). The building blocks of the AIs correspond to all isolated Wyckoff positions, x. These are, a, b, c, d, g, and contain all points reciprocal to the HSM; see Table V   VI. The complete K-theory classification of topologically distinct groundstates as well as the complete classification of topological phases with anomalous surface states, ASS = K/AI, of space-group P4. Each entry of the table is of the form K → ASS with K = KR −q,0 G k (S d k ) for AZ class, q, and HSM, k. The " · " symbol indicates a trivial classification, i.e., 0 → 0.

Overview of the full classification results
The complete K-theory classification, KRĜ(BZ), of topologically distinct groundstates as well as the the complete classification of topological phases with anomalous surface states, ASS = K/AI, of space-group P4 is presented in Table VI. In Sec. III we focused on the complete classification of anomalous surface states (see the P4 entries in Table II), these appear as the ASS in the last column of Table VI. In the following sections we derive these results in detail.

Topologically distinct groundstates
The complete K-theory classification, KR −q,0 G (T 3 ), of topologically distinct groundstates in AZ symmetry class, q, is derived from our free spectra decomposition, Eq. (21), and is given by our HSM decomposition, Eq. (22), i.e., a. Representation theory of Z 8 -SinceĜ Z 8 , we shall use the standard ungraded complex representations of Z 8 in order to label the Z 2 -graded representations in a prescribed manner. We denote the fundamental 1D complex representation of Z 8 by t 8 , such that Any representation of Z 8 is a polynomial with integer coefficients of t 8 subject to the relation (t 8 ) 8 = 1, such that each coefficient corresponds to the multiplicity of an irreducible representation, (t 8 ) n , and every representation has a complex conjugate representation ( For example, the 3D geometric representation in Eq. (37) is a real representation, t 2 8 + t 6 8 + t 4 8 , which is a direct sum of the 2π 4 -rotation 2D representation, t 2 8 + t 6 8 , and the 1D sign representation, t 4 8 , i.e., We use an analogous construction for the subgroup, C 2 ⊂ S 4 , withĉ 2 =ŝ 2 4 , such that t 4 = Res Z8 Z4 (t 8 ). b. Z 2 -graded representations -The Z 2 -grading itself, i.e., a group homomorphism,Ĝ k → Z 2 , is determined by the determinant of the O(d k ) geometric action ofĜ k on the primitive reciprocal lattice vectors, For example, the action in Eq. (37) yields the Z 2 -grading, In general, there are three types of possible gradings; we dub these, type-0, type-1, and type-2, as we now explain.
Type-0: This is the simplest type corresponding to the Γ, M -points, where the Z 2 -grading is trivial. In this case, the summand KRĜ k (S d k ) decomposes according to irreducible ungraded real representations ofĜ k . For example, consider the M -point, which is stabilized by the S 4 little group acting on b 1 , b 2 , such that the actions of all elements ofĜ M have positive Here, the complex K-theory summands correspond to irreducible real representations of complex type. Type-1: This is the type corresponding to the Z, X, R, A-points, where the Z 2 -grading is non-trivial. In this case, there is a canonical sign representation ρ sign , which stems from the Z 2 -grading; the summand KRĜ k (S d k ) decomposes according to Z 2 -graded representations ofĜ k , each of which has even and odd parts, ρ 0 and ρ 1 , with ρ 0 = ρ sign ⊗ ρ 1 . We encode this structure by the virtual representation, [ρ 0 −ρ 1 ]. For example, consider the R-point, which is stabilized by the C 2 little The Z 2 -grading,ĉ 2 ,ĉ 3 2 det − − → −1 and 1,ĉ 2 2 det − − → 1, provides us with a sign representation ρ sign = t 2 4 , and we thus have, where the shifts in the K-theory degree, i.e., (0, −1) and (−1, 0), are determined by the isomorphism of Ref. [62]; see Eq. (20).
Type-2: This type is only applicable for evendimensional irreducible representations of groups with a non-trivial Z 2 -grading. These are absent in cyclic groups but present in dihedral and other groups. In this case, these representations yield summands corresponding to (virtual) representations of the even part ofĜ k with respect to the Z 2 -grading. Examples are given in Ref. 62 and are not present in the classification of the current pedagogical example.
c. K-theory classification results -By repeating this analyses for all HSM we obtain, in Table VII, the complete decomposition of the equivariant K-theory classification, KR p,q G (T 3 ), into non-G-equivariant K-theory components for each irreducible Z 2 -graded representation. Note, that since we are using the double point group,Ĝ, we get contributions corresponding both to spinless and spinful electrons [131]. In order to distinguish these contributions, in Table VII, we place the spinless summands on the left and the spinful summands on the right. All summands may be easily read from the invariants in the periodic table of TISC (see Table I), using Bott periodicity [2,116,117], The spinful contributions for all AZ classes are explicitly presented in Table VI.
The complex AZ classes, A and AIII, are classified by complex K-theory which decomposes according to irreducible complex representations. In particular, every irreducible real representation of complex type splits into two irreducible complex representations, e.g., t 5 8 +t 3 8 splits to t 5 8 and t 3 8 . This is explicitly presented in Table VIII.

Anomalous surface states
The complete classification of topological phases with anomalous surface states in AZ symmetry class, q, is given by topologically distinct groundstates which are not related to one another by atomic insulators, i.e., ASS −q,0 = KR −q,0 G (T 3 )/AI −q,0 . In order to obtain these, one must quotient out the groundstates corresponding to AIs, i.e., the image of the ai map, AI −q,0 ai − → KR −q,0 G (T 3 ). As discussed in Sec. III B 1, the AIs themselves are described by the K-theory of a point for each isolated Wyckoff position, see Fig. 5 and Table V. Note, that the g Wyckoff position is a line of equivalent points and thus also classified by a single equivariant K-theory component. Similar to Tables VII and VIII, each of the Wyckoff position contributions is decomposed into non-G-equivariant components for each irreducible (ungraded) representation. This is presented in Tables IX and X. a. Structure of the ai map -The ai map is a Ktheory map. Specifically, it is a homomorphism between the modules, AI * , * and KR * , * G (T 3 ), over the ring, KR * , * G (pt), i.e., a map which respects the multiplicative structure of K-theory, Eq. (29). As discussed in Sec. III B 4, such maps satisfy, for all u ∈ AI p,q . Since 1 ∈ KR 0,0 G (pt) is a spinless representation, it is now clear why we have not omitted these contributions in Tables VII-X. The ai * , * map is thus completely determined by ai 0,0 (1). Furthermore, it follows from Eq. (22) and Eq. (47) that the ai map can be displayed as a blockmatrix, [ai] kx = [ai(u x )] k , whose entries are the con- . For each k and x, the block's entries correspond to the irreducible Z 2 -graded representations of G k and the irreducible representations ofĜ x as given in Tables VII and IX. We where 1 is the identity element, 1 ∈ KR 0,0 (pt) 1 . Note, that KR p+pρ,q+qρ (S d ) ρ=[ρ 0 −ρ 1 ] is a shifted copy of KR p,q (pt), therefore, we denote its canonical generator by We use a similar construction for KC components. Note, that none of the maps discussed in Sec. IV A are affected by the invertible elements, β, µ, and ξ 4 . Therefore, here and henceforth, we set µ = β = ξ 4 = 1 for all elements of KR p,q (S d ) = KR p,q−d (pt), see, e.g., Eq. (56). Let us explain how to obtain the matrix in Eq. (49). We begin by noting three generic properties: First, one notices that [ai] kx is an upper triangular block-matrix. The component KRĜ k (S d k ) does not capture groundstates which are topologically equivalent to states on the boundaries of the cell centered at k, see Fig. 4. This provides us with a partial order where ev-ery HSM is only contributed from 'greater' Wyckoff positions, Table V, are greater than those of k 0 . Hence, an AI at a Wyckoff position, x 1 , reciprocal to k 1 , only contributes to KRĜ k 0 Second, one notices that all entries left to the diagonal are multiples (within the representation ring) of the diagonal entry. Consider a particular diagonal entry with k 0 reciprocal to x 0 . The contribution to KRĜ k 0 (S d k 0 ) of an AI at any x 1 > x 0 is given by restricting the groundstate to the sub-torus centered at k 0 . This gives an AI with the same k-dependence as that at x 0 and with a representation determined by an appropriate restriction and/or induction betweenĜ x1 andĜ x0 . For example, consider an AI at the g Wyckoff position with a trivial representation ofĜ Z 4 . It corresponds to two s- , which are interchanged byŝ 4 . Therefore, as a representation ofĜ Γ Z 8 , its con- , which is a two-dimensional representation corresponding to both orbitals combined.
Third, one notices that no Wyckoff position contributes to the R-point. This is a consequence of the high-symmetry line that connects the R-point with the X-point; see Fig. 5. Any AI groundstate supported on the g Wyckoff position, x = ( 1 2 , 0, x), (0, 1 2 , x), may be continuously deformed to x = ( 1 2 , 0, 0), (0, 1 2 , 0) which does not contribute to the R-point. This is also a generic property.
An immediate conclusion of these three properties is that the [ai] kx block-matrix may be block-diagonalized using column operations [valued in Rep(Ĝ)]. This implies that the image of the AI map is identical to the image of the block-diagonal entries. These entries correspond to contributions of AIs at the center of the point-group and thus may be evaluated using Shiozaki's formula [127], Eq. (28).
b. Application of Shiozaki's formula -In order to interpret Eq. (28) in terms of our decomposition, we utilize the Atiyah-Bott-Shapiro construction [116,132] which expresses elements of the real K-theory in terms of Z 2 -graded representations of Clifford algebras, Cl p,q . In essence, this amounts to lifting the geometric action, g., Eq. (37)], to a "geometric-algebra" actionG k → Cl d k ,0 and decompose this action into irreducible Z 2 -graded representations. We leave the mathematical details to Appendix B 6 and provide here a "quick and dirty" method to obtain the diagonal entries of [ai 0,0 (1)] kx .
The gist of the "quick and dirty" method is as follows.
We first use the geometric action, and restrict our attention to the image, . We then construct a double cover,G k , of G k , which is faithfully represented by SU (2), which itself is the double cover of SO(3), i.e., such thatG k ⊂ SU(2) [133]. The unitary matrices U g ∈ SU(2) are constructed as to satisfy, where σ 1,2,3 are the Pauli matrices. Finally, the diagonal entry of the ai 0,0 (1) matrix [Eq. (49)] is simply found by decomposing the above unitary matrix representation into irreducible ungraded representations ofĜ k .
Note, that the faithful representation, Eq. (53), ofG k provides us with a generically non-faithful representation ofG k G k . Here,G k is a double cover of G k which might be different than the physical double point-group, G k . Fortunately, for all HSM of space-group P4, one has G k Ĝ k (the generic case is treated in Appendix B 5). One might worry that some data is lost when restrict- Table. VII), we at most encounter a global sign ambiguity for each row of [ai 0,0 (1)] kx ; this does not alter the quotient by the image of the ai map. One may always get rid of this inconsequential ambiguity by following Appendix B 6.
Let us provide a couple of explicit examples, which would make our method clearer: First, consider the contribution of an AI at the c Wyckoff position to the M -point. The This is a faithful action, so G M = G M Z 4 . We construct a faithful unitary matrix representation ofG M = G M Z 8 , given by, This is clearly the t 8 + t 7 8 representation, which leads to [ai 0,0 (1 c )] M = [t 8 + t 7 8 ]ξ 3 in Eq. (49). The ξ 3 factor stems from where we have used µ = ξ 4 = 1 as above.
Second, consider the contribution of an AI at the b Wyckoff position to the Z-point.
This is not a faithful action, so we restrict to G Z Z 2 . We construct a faithful unitary matrix representation of G Z Z 4 , given by, Here, in order to spare the reader of excess notation, we identified the generator ofG Z with theŝ 4 generator of . This is where the inconsequential global sign ambiguity comes about.
c. Application of K-theory multiplication -Once , for all u ∈ AI −q,0 and all AZ symmetry classes, q. This provides the image of the ai map, that is, the full classification of AI groundstates and thus ASS −q,0 = KR −q,0 It thus suffices to find the images of the generators of AI −q,0 , as given in Table IX, corresponding to the irreducible representations ofĜ x .
We thus focus on each block-diagonal entry of [ai] kx and expand that block into a matrix [ai] ρ k ρx = [ai(u ρx )] ρ k whose entries are the contributions of the AI, u ρx ∈ KR p,q Gx (pt), to the ρ k -component of KR p,q G k (S d k ). Each ρ k and ρ x correspond to irreducible Z 2 -graded representations ofĜ k and the irreducible representations ofĜ x as given in Tables VII and IX. We find that the blocks, [ai] ρ k ρx = [ai(u ρx )] ρ k , for space-group P4 are as follows: [ai] Γa is always the identity matrix, [ai] Rg is the zero matrix, and the rest are given by Here, we have only presented the results for the spinful components in Tables VII and IX; the spinless results are analogous.
In order to obtain these results, we utilize the multiplicative structure of the equivariant K-theory. In particular, one finds that the inherent equivariant spectra constrain the ai map to such an extent that it is uniquely determined by the multiplicative structure of the ungraded complex representations, as we immediately demonstrate using an explicit example. Consider within cf. Eq. (56). Following the discussion in Sec. IV A, the only possible K-theory map, stemming from a modulespectra homomorphism of type KC → KR ⊕ KR, is given by the realification map, where n , n ∈ Z are integers. Moreover, from the relevant entry of Eq. (49), we know that In order to extract the integers, n and n , we observe that Eq. (66) translates to the complex K-theory in Tables VIII and X as follows, The complex K-theory maps are determined by the multiplication rules of the ungraded complex representations of Z 8 . Specifically, we recall that ai(u) = u · ai(1), and thus by setting ξ = 1, we may utilize to conclude that n = n = 1. All other entries of Eqs. (59)- (63) are similarly obtained using the correspondences between the real and complex K-theories which are provided in Table XVI in Appendix B. Note, that since r(u) = η for all u ∈ KC −q,0 (pt), the only type of K-theory maps undetermined by the complex representation theory is v → nη · v. Nevertheless, by 2η = 0, one has nη ·v = (n mod 2)η ·v, such that this map is completely determined by the dimensionality (modulo 2) of the representation. d. Classification of anomalous surface states -Once in full possession of ai(u), for all u ∈ AI −q,0 , it is but a simple manner of picking a particular AZ symmetry class, q, and quotienting-out the image of the ai map, AI −q,0 ai − → KR −q,0 G (T 3 ), in order to finally obtain all anomalous surface states ASS −q,0 = KR −q,0 G (T 3 )/AI −q,0 . As discussed above, the complete classification results are presented in Table VI. In order to demonstrate the abelian calculations leading to it, let us focus on the Apoint HSM in AZ symmetry class AII of q = 4, where For any particular AZ class, ai −q,0 is an abelian-group homomorphism. For our case, it takes the form, This is obtained using Tables VII, IX, and Table XVI in Appendix B, by focusing on spinful-electrons; specifically, cf. Eq. (56). Using our explicit results in Eq. (63), and setting u = ξ 2 = ξ −2 (by ξ 4 = 1 as discussed above), we find that the [ai] Ad matrix is given by Here, we have evaluated the realification map as given in Table III.
In general, in order to find the quotient by the image of an integer-valued matrix, i.e., its cokernel, one brings it to its Smith normal form, where the elementary divisors are the diagonal entries. The [ai] kx matrix is KR * , * (pt)-valued; nevertheless, we may always manually impose 2η = 2η 2 = 0 and bring it to a pure integral formulation. In our case, this amounts to adding an extra 0 2η 2 -column. We find, Finally, since Z 1 × Z 4 Z 4 , we conclude that In particular, we find that if we stack four copies of the elementary topologically distinct groundstate of the [1 − t 4 8 ] A representation we get a state which is topologically equivalent to an AI. All other invariants in Table VI or obtained via analogous calculations.

Summary
This completes our "top to bottom" derivation of the full classification of topologically distinct groundstates and topological phases with anomalous surface states for space-group P4. We have constructed the following data: (i) The identification of the HSM with π-integer primitive coordinates, their reciprocal Wyckoff positions, and their stabilizer double point-groups (little groups); see Table V.
(ii) The decomposition of the full K-theory classification into components corresponding to each HSM via our equivariant spectra paradigm; see Eq. (38).
(iii) The decomposition of the K-theory component corresponding to each HSM into Z 2 -graded representations; see Tables VII and VIII. This yields the complete classification of topologically distinct groundstates; see Table VI. This yields the complete classification of topological phases with anomalous surface states; see Table VI.
Such calculations were implemented as a GAP4 [129] language algorithm and yielded all the topological classification tables in Table II and Appendix A. A much deeper mathematical perspective is provided in Appendix B but the gist of all crucial steps have been explicitly demonstrated. Before concluding our paper in Sec. V, we depart from the theoretical algebraic perspective taken so far, which have been extremely useful in obtaining quantitative results, and use our test-case of space-group P4 to provide the reader with tight-binding Hamiltonians manifesting our predicted anomalous surface states.

C. Model Hamiltonians
The constructive isomorphism in Eq. (20) provides explicit model Hamiltonians for all classes of topologically  Table VI. This is depicted in Fig. 6.
With this choice of matrices, the Hamiltonian [Eq. (77)] exhibits topologically protected anomalous surface states and topological phase transitions for various values of µ. These surface states are clearly visible in Fig. 6 where we aesthetically set t = t, ∆ = ∆ = 2t, and µ = 2t. The Dirac cone is fourfold degenerate and is topologically protected by Θ, Ξ, andŝ 4 .
As discussed in Sec. III B 4, the topology of this chiral superconductor cannot be captured by the symmetry indicators, as there are no symmetry indicators for gapped systems in AZ class DIII. Moreover, it is straightforward to check that it has a zero winding number and hence also eludes the Z winding number invariant of 3D chiral TSCs in AZ class DIII; see Table I. This is also evident without explicit calculation by noting that the only possible homomorphism from the Z 2 classification of CTISC we found to the Z winding number of TISC, Z 2 → Z, is the zero homomorphism.
Finally, we emphasize that there is nothing unique about the choice of AZ class DIII, and the same procedure of constructing Dirac matrices and tight-binding Hamiltonians may be applied to all ten AZ symmetry classes.

V. DISCUSSION AND OUTLOOK
A. Comparison with other works

Topologically distinct groundstates
We have obtained unified results for the classification of topologically distinct groundstates for the full tenfoldway of the two complex and eight real AZ non-spatial symmetry classes; see Fig. 1 In general, the E ∞ -page of the Atiyah-Hirzebruch spectral sequence approximates the K-theory. Our K-theory results consistently fit their spectral sequence results.

Topological phases with anomalous surface states
One of our main results, see Table II, is the unified classification of topological phases with anomalous surface states for the full tenfold-way of all AZ symmetry classes, A, AIII, AI, BDI, D, DIII, AII, CII, C, and CI; see  Fig. 4), analogous to strong TISC in the non-crystalline case. These results for magnetic point-group symmetries may be compared with our results for AZ classes, A, AI, BDI, D, DIII, and AII, and are in complete agreement.

Symmetry indicators
The majority of SI studies have naturally focused on Dyson's threefold-way of AZ classes A, AI, and AII [14,125]; see, e.g., Refs. 65 and 66. Nevertheless, an extension to the other AZ classes has been successfully achieved; see, e.g., Refs. 72, 75-77.
However, as discussed in Sec. II, the anomalous surface states do not necessarily have to be indicated by the SI; see Fig. 2. Moreover, it is possible for SI to indicate a gapless state such as a (semi-)metal. Nonetheless, we find that whenever the SI indicate gapped states (i.e., CTISC) we find them to be quotients of our K-theory classification as required by Eq. (1).
In particular, a direct non-trivial extension of this work may be in the study of invertible symmetry protected topological phases of strongly-interacting fermions which are classified by real cobordisms [26].
There is an intriguing sense in which one may gradually interpolate between K-theory and real cobordisms, by mean of the chromatic filtration [135]. Accordingly, it is natural to hypothesize that the topological phases of fermions with higher-order interactions (e.g., quartic interactions), are classified by cohomology theories of higher chromatic heights. We unfortunately as yet have no evidence for such a relation, other than the two extreme cases.

Magnetic space-groups & topological superconductivity
In this paper we have limited our scope to symmetry actions where the spatial symmetries all commute with the anti-unitary non-spatial symmetries, Θ and Ξ, see Fig. 1. However, although more complicated, our analysis is generalizable to treat cases where these symmetries intertwine [136].
Recently, in Refs. 75-77, Geier et al. and Ono et al. have studied the SI of superconducting systems where they had also treated the possibility of intertwining the particle-hole anti-symmetry, Ξ, and the spatial symmetries. Since Θ and Ξ are both manifestations of Z T 2 with different actions on the underlying KU spectrum, both intertwinings should be akinly treated in our paradigm; see discussion in Sec. II B 1.
In Refs. 89 and 90, Trifunovic, Brouwer, and Geier have presented a theoretical formulation for the classification of such HOTISC using a filtration of the K-theory. In Ref. 98, Shiozaki, Xiong, and Gomi have shown that this filtration naturally fits within the Atiyah-Hirzebruch spectral sequence. We expect that the knowledge of the full K-theory classification provided in this paper would be used to resolve the sequence and obtain a complete classification of HOTISC phenomena.

Hexagonal space-groups
While the general machinery used in this paper is relevant also for the hexagonal space-groups, one key ingredient breaks down. Namely, the free G-equivariant spectrum of a torus with a hexagonal action does not decompose into a direct sum according to its cells, and so the G-equivariant K-theory does not reduce to equivariant spheres. Nevertheless, there are still advantages in the equivariant spectra paradigm.
IfT d is a torus with an action of a group G, one can always equivariantly constructT d out of cells [63,97]; cf. Fig. 4. For example, consider layer-group p6mm, where C 6v acts on the 2D torus,T 2 . This resulting Gequivariant torus may be decomposed into two equilateral triangles, permuted transitively by the C 6v -action. While the G-equivariant attaching data of the triangles to their boundary no longer stably trivializes, the part of the data determining the G-equivariant K-theory of the torus is computable and depends only on a small number of parameters. Hence, we expect that our methods will prove advantageous in the classification of topological phases for hexagonal symmetry as well.

C. Summary
In this paper, we have utilized the mathematical equivariant spectra paradigm to obtain explicit quantitative results for the classification of topologically distinct groundstates as well as topological phases with anomalous surface states of crystalline topological insulators and superconductors. This is done in a unified manner, which captures the full tenfold-way of Altland-Zirnbauer non-spatial symmetry classes; see Fig. 1. In Table II, we have focused on the full classification of key 3D spacegroups, but our analysis naturally captures 2D layergroups and 1D rod-groups as well; see Appendix A. These full classification results are exhaustive and thus extend beyond the symmetry indicators of band topology.
We have established both the theoretical and the computational benefits of the equivariant spectra paradigm: First, we have successfully utilized the modern mathematical framework of equivariant spectra to obtain a geometric equivariant spectra formulation of crystalline systems; see Eq. (21). These infrastructural spectra are independent of the specific classification of topological phases we set out to complete.
Next, we emphasized the K-theory multiplication which physically translates to the hidden multiplicative Space-group · · · HSM · · · . . .

AZ class
K → ASS . . . structure within the periodic table of topological insulators and superconductors, mixing the different AZ symmetry classes. This fundamental multiplicative structure is independent of any crystalline symmetry. Nevertheless, this generalizes to the crystalline case and unveils further relations between different CTISC phases; see Sec. III B 4.
Furthermore, by treating the atomic insulators and superconductors within the same equivariant spectra paradigm, we provide a deeper understanding of the AIs' classification and thus of the anomalous surface states. This understanding translates to an efficient computational approach [see Sec. IV B 5] which allowed us to attain results for the full tenfold-way of AZ symmetry classes in a unified manner.
To conclude, as discussed above, we showed that our results consistently broaden the existing knowledge of CTISC phases and that the paradigm we have established holds the potential to lead the path towards the discovery and the understanding of other diverse topological phenomena.

ACKNOWLEDGMENTS
We are grateful for illuminating discussions with A. Nagy, N. Okuma, K. Shiozaki, R. Thorngren, E. Berg, D. Clausen, and T. Schlank. E. C. acknowledges support from CRC 183 of the Deutsche Forschungsgemeinschaft. S. C. was supported by the Adams Fellowship Program of the Israel Academy of Sciences and Humanities.

Appendix A: Classification tables
In this appendix, we present the full classification tables which provide detailed information that complements the main results presented in Table II.
In Table XII (ahead) we present the complete K-theory classification of topologically distinct groundstates for the signed-permutation representation space-groups as well as the complete classification of topological phases with anomalous surface states, ASS = K/AI. Each entry of the table is laid-out as in Table XI which  The abelian groups presented in Table XII may be easily used to obtain the full classification of many layer-groups and rod-groups as well. Since any layergroup/rod-group is a sub-group of some space-group, it is straightforward to extract their topological classification. This is discussed in Sec. A 1.  Table II, we provide the complete classification of topologically distinct groundstates as well as the complete classification of topological phases with anomalous surface states. Each entry of the table is of the form K → ASS; see discussion in Appendix A. The " · " symbol indicates a trivial classification, i.e., 0 → 0.

Layer-groups & rod-groups
The BZ of either a layer-group or a rod-group system is correspondingly either a 2D or a 1D sub-torus of some embedding 3D space-group. Therefore its topological classification is a direct sum of the topological invariants corresponding only to the HSM of the sub-torus.
As an example, consider space-group P4 discussed in Sec. IV B. It's topological classification is obtained as a direct sum over the Γ, Z, X, R, M, A-points. The layergroup p4 is a subgroup of P4 and its 2D BZ contains only the Γ, X, M -points; see Fig. 5. The direct sum of the topological invariants of these HSM, which are tabulated in Table XII, provide the complete classification for layergroup p4. Similarly, the rod-group p4 is a subgroup of P4 and its 1D BZ contains only the Γ-point and the Z-point; see Fig. 5. The direct sum of the topological invariants of these HSM provide the complete classification for rodgroup p4. This is explicitly demonstrated in Table XIII.

Appendix B: Mathematical background
The main objects of study in this paper is the Gequivariant real K-theory KR p,q G (T ) of a torusT and the image of the AI-map ai p,q : AI p,q → KR p,q G (T ) in it; see Sec. III B. For every value of (p, q), the map ai p,q is a map of abelian groups, and it is not so easy to compute these maps independently. One of the insights that we present is that these maps are highly interlaced. They are all different manifestations of a single map of objects, which depend on a much smaller number of parameters then a tuple of maps of abelian groups for every value of (p, q) ∈ Z 2 . To explain this common origin and dependencies between the various degrees, let us first give a "shadow" of it on the level of cohomology groups.
This fact already provides many relations between the maps, ai p,q , for various (p, q)-s. As a first trivial application, using the multiplication by µ and β, we see that the map depends only on its (p, 0) component for 0 ≤ p ≤ 7, recovering the classical observation that the computation of the entire ai map reduces to those degrees. However, and especially for summands of AI * , * of the form KC * , * (pt), this is not enough to entirely determine the map ai. Specifically, the module KC * , * (pt) is not a free module over KR * , * (pt), and is not generated by a single element, but rather it is minimally generated by 1, ξ, ξ 2 , and ξ 3 . This deficiency naturally leads to the question of whether every map of KR * , * (pt)-modules, i.e., a map that respects the multiplication by classes of the real Bott-periodicity, can appear as a matrix entry in a matrix representation of ai * , * . For example, one can ask whether every map KC * , * (pt) → KR * , * (pt) can appear in the computation. It turns out that this is not the case. For example, the map given by cannot appear as such a matrix entry, even though it is entirely consistent with the KR * , * -module structure. The reason for this phenomenon is that the map ai must come from a map of modules over the ring-spectrum KR itself, not just the cohomology ring KR * , * (pt). Namely, the map ai intertwines the operation of tensoring with a real vector space on the geometric level, before passing to homotopy groups. To make this idea precise, one needs to explain in what sense, and where, the object KR is a ring, the objects AI and KR G (T ) are modules over KR and ai : AI → KR G (T ) is a map of modules over KR. A natural context to make these notions precise is that of Z T 2 -equivariant spectra. It turns out that thinking about the classification problem in terms of equivariant spectra has many other advantages that we shall later explain.
The rest of this appendix is organized as follows: In Sec. B 1, we recall the classical theory of rings, modules, and representations of groups, both in the usual and the Z 2 -graded settings. In Sec. B 2, we discuss real and complex K-theory, their multiplicative structure, and the relations between them. In Sec. B 3, we consider the notions of spectra and G-equivariant spectra. We discuss the tensor product of spectra and the resulting theory of commutative rings and modules internally to spectra. We then specialize to the case of G-equivariant real K-theory and explain how to define it as a commutative ring in G-equivariant spectra. In sec. B 4, we study the G-equivariant K-theory of the torus associated with a special type of equivariant lattices which we call signed permutation representations. We show that the G-equivariant spectrum generated from such a torus splits as a direct sum of representation spheres, hence reducing our calculations to the case of spheres. In Sec. B 5, we study the G-equivariant K-theory of representation spheres endowed with a Pin-structure. We show how a Pin structure on the representation allows us to compute the G-equivariant K-theory of the representation sphere as a real K-theory module using only representationtheoretic considerations. Finally, in Sec. B 6, we use all the machinery introduced in the previous sections, and Shiozaki's formula, to give a uniform classification, in representation-theoretic terms, of the atomic insulators for crystals with signed permutation representation symmetry.

Rings, representations, & gradings
In the next sections, we shall discuss some "brave new algebra", i.e., algebra done in the category of spectra. Hence, as a warm-up, we recall some classical notions from classical algebra.

a. Rings & modules
Recall that a ring is a set R with two binary associative operations + and ·, such that + makes R into an additive abelian group and the distributive laws hold: (a + b)c = ac + bc and a(b + c) = ab + ac. We also demand the existence of a unit for the multiplication, 1 ∈ R, such that 1a = a1 = a. A typical example is the ring of real n by n matrices, M n×n (R) with addition and matrix multiplication. Note, that we do not require the multiplication to be commutative. If it does, i.e., if ab = ba, we say that R is a commutative ring. A typical example is the commutative ring Z of integers with addition and multiplication.
If R is a ring, we can form the ring of polynomials, with its natural addition and multiplication of polynomials. More generally, if R is a commutative ring and f 1 (x 1 , .., x ), . . . , f m (x 1 , . . . , x ) are polynomials with coefficients in R, we can form the ring, given by adding formal variables x i subject only to the relations, f j (x 1 , . . . , x ) = 0, for j = 1, . . . , m. This will be a common way for us to present the rings involved in our computations. For example, in this convention, we have C R[x]/(x 2 + 1). In fact, the ring in Eq. (B3) is an R-algebra, i.e., endowed with a ring map from R. More precisely, it is the R-algebra with generators x i and relations f j (x 1 , . . . , x ).
A module over a ring R is an abelian group M on which R acts linearly. More precisely, we have a binary operation, (r, m) → rm, which satisfies (rr ) · m = r · (r · m), (r + r ) · m = r · m + r · m and r · (m + m ) = r · m + r · m . Finally, we demand that 1 · m = m. Equivalently, if M is an abelian group, the set of homomorphisms, Hom Ab (M, M ), of abelian groups from M to M admits a ring structure, for which the addition and multiplication are given by (f + g)(m) = f (m) + g(m) and (f g)(m) = f (g(m)), the multiplicative unit is the identity map Id M . An R-module structure on M is just a ring homomorphism R → Hom Ab (M, M ). The collection of modules over R organizes into a category Mod R (Ab) of R-modules in abelian groups. This means that for every two R-modules M, M there is a set of R-linear maps, Hom R (M, M ), i.e., group homomorphisms for which f (r · m) = r · f (m). In fact, Hom R (M, M ) has a canonical addition turning it into an abelian group. If R is moreover commutative, then Hom R (M, M ) admits an R-module structure, via (r · f )(m) = r · f (m) = f (r · m), and we always consider Hom R (M, M ) as endowed with this module structure. In the case where M = M we also denote Hom R (M, M ) by End R (M ).

b. Group rings & representations
If G is a group and R is a ring with a G-action given by r g − → g(r) ∈ R, one can produce the twisted group ring, R[G], given by formal sums, i r i g i , with addition, rg + r g = (r + r )g, and multiplication, rg · r g = (r · g(r ))(gg ). (B4) In the special case where G acts trivially, so that g(r) = r, we obtain the (non-twisted) group ring, R[G].
The seemingly arbitrary construction of the twisted group ring is characterized up to Morita equivalence via a very simple property of the category of modules. Since the ingredients involved in the characterization play a crucial role in the passage from groups and algebras to spectra and ring spectra, we shall discuss it before moving on to graded objects. Let Ab G denote the category of abelian groups with G-action. We can think of R as a ring in Ab G , i.e., a ring with a G-action. More precisely, groups with G-action have a tensor product, (M, N ) → M ⊗ N , given by the usual tensor product with the G-action g(m ⊗ n) = g(m) ⊗ g(n). Then, the action of G on the ring R is encoded via a multiplication, R ⊗ R → R, where both the source and target are in Ab G , and the tensor product is the tensor product in Ab G . Since R is a ring in Ab G , it makes sense to talk about modules over R internally to Ab G . We denote this category by Mod R (Ab G ). Explicitly, this is an object, M ∈ Ab G , together with action map, R ⊗ M → M , satisfying the usual module identities. We have an equivalence of categories In the special case where R = R or R = C with the trivial G-action, we recover the usual group rings R To describe this isomorphism in a canonical way, we shall discuss the notion of group representation in more detail. If G is a finite group, a (complex) representation of G is a complex vector space U and a group homomorphism, ρ : G → GL(U ), where GL(U ) is the group of invertible linear maps from U to U . It is always possible to choose a Hermitian metric on U which is preserved by the Gaction. By choosing such a metric and an orthonormal basis to U , we can depict a d-dimensional representation as a group homomorphism ρ : Let ρ 1 , . . . , ρ be the set of pairwise non-isomorphic irreducible representations of G. The map ρ i , given in some orthonormal basis, by ρ i : G → U(d i ) ⊆ M di×di (C), extends by linearity to a map, C[G] → M di×di (C). The isomorphism in Eq. (B6) is then the direct sum of the these extended maps.
In the real case, the situation is slightly more complicated, as real irreducible representations naturally split into 3 kinds. To describe those, we shall consider some standard linear algebra constructions related to real and complex vector spaces.
A real vector space, V , has a complexification, c(V ) Similarly, a complex vector space, U , has a realification, r(U ) U , considered as a real vector space by forgetting the multiplication with complex scalars. Accordingly, a real representation, τ : G → GL(V ), can be complexified, so that and a complex representation, ρ : G → GL(U ), can be realified via Finally, a complex vector space U has a complex con-jugateŪ , which is the same real vector space with the scalar multiplication defined by a · v def =āv. Moreover, this construction is functorial, i.e., a complex-linear map, A : U 1 → U 2 , induces a linear map,Ā : U 1 → U 2 . Accordingly, a complex representation, ρ : G → GL(U ), has a complex conjugate, In coordinates, the mapρ : The operations, c and r, satisfy the identities, c(r(U )) U ⊕Ū and r(c(V )) V ⊕ V .
Let τ : G → GL(V ) be an irreducible real representation. The representation, ρ def = c(τ ), is equipped with an involutive isomorphism, κ :ρ ∼ − → ρ, coming from the complex conjugation on C. In fact, a real representation can be identified with a complex representation endowed with such an involutive isomorphism with the complex conjugate. If c(τ ) decomposes as a direct sum of two different irreducible representations, c(τ ) ρ 1 ⊕ ρ 2 , then End G (τ ) C, and τ contributes a summand of the form M d×d (C) to R[G]. We say in this case that τ is of complex type. Moreover, in this case ρ 2 ρ 1 and r(ρ 1 ) r(ρ 2 ) τ .
If ρ = c(τ ) is an irreducible complex representation, then the involutive isomorphism, κ :ρ ∼ → ρ, endows ρ with a real structure, and so τ contributes a summand of the form M d×d (R) to R[G], with d = dim(τ ). In this case, we say that τ is of real type, and one has r(ρ) τ ⊕ τ .
Finally, it is possible that c(τ ) ρ ⊕ ρ for a single complex irreducible representation, ρ. In this case we have an isomorphism, κ :ρ ∼ → ρ, but we can not choose κ to be involutive. Rather, the composition, ρκ − →ρ κ − → ρ, can be chosen to be the scalar linear map, − Id ρ . Then, the representation τ contributes a copy of M d/4×d/4 (H) to R[G], where d = dim(τ ) and H is the quaternion algebra, i.e., the R-algebra generated by i and j with i 2 = −1, j 2 = −1, and ij = −ji. We say that τ is of quaternionic type; one has r(ρ) τ and c(τ ) ρ ⊕ ρ.
To conclude, we have an isomorphism, Representations of different groups are related. If H ⊆ G is a subgroup and τ is a real, or complex, representation of G, we can regard it as a representation of H, which we denote by Res Similarly, we denote by RRep(G) the Grothendieck ring of real representations of G. There is a ring homomorphism, c : RRep(G) → CRep(G), turning CRep(G) into an RRep(G)-algebra. In the opposite direction, we have a map, r : CRep(G) → RRep(G), of RRep(G)modules. Namely, for u ∈ CRep(G) and v ∈ RRep(G) we have Moreover, since the complexification is a ring homomorphism, one has Importantly, the realification is not a ring homomorphism but rather a module homomorphism and hence no analogous relation exists for r, cf. Eq. (B12). Furthermore, the ring CRep(G) admits an involution, u →ū, reflecting the complex conjugation of complex representations. The operations, c(v), r(u) andū, satisfy the relations, Computations within the G-equivariant K-theory of spaces requires not only representations and rings but also with their Z 2 -graded versions.
A Z 2 -graded abelian group is an abelian group, A, with a decomposition, A = A 0 ⊕ A 1 . We call A 0 the even part of A and A 1 the odd part of A. We let Ab Z2 denote the category of Z 2 -graded abelian groups. For A ∈ Ab Z2 , we say that a ∈ A is homogeneous if either a ∈ A 0 or a ∈ A 1 . In this case the parity of a is simply We also have a tensor product, A⊗B, on Ab Z2 given by Accordingly, we can define Z 2 -graded rings and modules over such. A Z 2 -graded ring is a Z 2 -graded abelian group R endowed with an associative and distributive multiplication, R⊗R → R, as well as a unit, 1 ∈ R 0 . We say that R is super-commutative if ab = (−1) |a||b| ba for homogeneous a, b ∈ R. We denote the category of Z 2 -graded modules over R by Mod R (Ab Z2 ). Specifically, an object, M ∈ Mod R (Ab Z2 ), is a Z 2 -graded abelian group M endowed with an associative and distributive action, R⊗M → M . One way to produce, not necessarily supercommutative, Z 2 -graded rings, is as group rings of Z 2 -graded groups. Here, a Z 2 -graded group is a group, G, endowed with a homomorphism, : G → Z 2 . We then set G 0 def = −1 (0) and G 1 def = −1 (1), such that G 0 is the even subgroup and G 1 is the odd part which is not a group. If G is a Z 2 -graded group, a Z 2 -graded (complex or real) representation of G is a (complex or real) representation, ρ : G → GL(U ), for a Z 2 -graded TABLE XIV. Real Clifford algebras, Clp,q. The even part of each algebra is isomorphic to Clp−1,q found one row above it. The first few algebras are explicitly displayed, all others are obtained via Bott-periodicity, Clp+1,q+1 M2×2(Clp,q) and Clp+4,q Clp,q+4 M2×2(H) ⊗ Clp,q.
The Z 2 -graded complex representations of G are encoded in the Z 2 -graded group ring, ] (note that the latter is not a group ring). Similarly, we have , factors into a sum of simple Z 2 -graded algebras according to the classification of the irreducible complex Z 2 -graded representations. In order to provide this factorization, we recall the notion of Clifford algebras.

d. The Clifford algebras
For p ∈ N, we denote by Cl p the Clifford algebra on p generators. Namely, Cl p is the Z 2 -graded algebra over C generated by the odd elements, e 1 , . . . , e p , with e 2 i = −1, and e i e j = −e j e i for i = j. Similarly, for p, q ∈ N, the real Clifford algebras, Cl p,q , are generated by the elements, e 1 , . . . , e p , e p+1 , . . . , e p+q , subject to the relations, and such that that the generators are all odd, i.e., |e i | = 1.
Particularly, one has Cl p+q = C ⊗ R Cl p,q . The first few real Clifford algebras are presented in Table XIV. Let G be a group which is Z 2 -graded by : G → Z 2 . The Z 2 -graded algebra C [G] decomposes as a direct sum of Z 2 -graded algebras of the form M d×d (Cl p ). Similarly, The Z 2 -graded algebra R [G] decomposes as a direct sum of Z 2 -graded algebras of the forms, M d×d (Cl p ) and M d×d (Cl p,q ). This decomposition is tied with the vital notion of Z 2 -graded representations which we now explain. e. Z2-graded representations Let us begin with complex representations. First, if is trivial, so that G 0 = G, then C [G] is entirely even, and the decomposition coincides with the one in Eq. (B6).
Otherwise, the construction, U → U 0 , induces an identification [139], with inverse given by Ind G G0 (U ). Hence, irreducible Z 2 -graded representations of G are in direct correspondence with irreducible representations of G 0 . Let us fix an arbitrary odd element, g ∈ G 1 . If τ is an irreducible representation of G 0 , so that ρ = Ind G G 0 (τ ) is an irreducible Z 2 -graded representation of G, we can use g to twist the representation τ and obtain a new representation, τ g , such that τ g (h) def = τ (ghg −1 ). The relation between τ and τ g determines the contribution of τ to C [G]. As with the classification of real representations, there are two types of irreducible representations of G 0 .
If τ g τ , we necessarily have Ind G G 0 (τ ) ρ 1 ⊕ ρ 2 for two different irreducible representations, ρ 1 and ρ 2 , of G. In this case, the Z 2 -graded representation ρ contributes a copy of M d×d (Cl 1 ) to C [G] for d = dim τ . The underlying ungraded C-algebra is then isomorphic to M d×d (C) ⊕ M d×d (C) and corresponds to the summands of C[G] coming from ρ 1 and ρ 2 .
If τ and τ g are different irreducible representations, then Ind G G 0 (τ ) is irreducible as an ungraded representation. In this case, as ungraded representations we have Ind G G 0 (τ ) Ind G G 0 (τ g ) ρ and so, there are two nonequivalent Z 2 -gradings on ρ. In this case, the pair (τ, τ g ) contributes a copy of M d×d (Cl 2 ) to C [G]. The underlying ungraded algebra is M 2d×2d (C), which is the summand of C[G] corresponding to ρ.
To conclude, if : G → Z 2 is non-trivial, we have Moving on to real representations, one finds that the decomposition of a Z 2 -graded group ring R [G] into real Clifford algebras is more complicated as it involves both the complex conjugation and the twisting by g ∈ G 1 . First, if G 0 = G, then R [G] is entirely even, and the decomposition coincides with the one in Eq. (B10). Otherwise, there are ten different types of real Z 2 -graded representations which are summarized in Table XV. These correspond to the ten AZ symmetry classes.

Real and complex K-theory
a. The ring structure on KU Complex K-theory is a classical generalized cohomology theory introduced by Atiyha and Hirzebruch [140], TABLE XV. The ten irreducible Z2-graded representations and their corresponding matrix algebras. For each irreducible ungraded representation τ of G 0 , there is a Z2-graded representation, ρ = Ind G G 0 (τ ), of G. It is determined by the real, complex, and quaternionic structures on τ and ρ, and on whether τ and τ g are isomorphic or not for g ∈ G 1 .
see also Ref. 117. Recall that a vector bundle on a topological space X is a choice of a finite dimensional complex vector space U x for every x ∈ X, depending continuously, in an appropriate sense, on x ∈ X. We let KU 0 (X) be the abelian group generated by all complex vector bundle on X, subject to the relation, Every element in KU 0 (X) can be written as a formal difference, , which we refer to as a virtual bundle, i.e., an hypothetical object that, if added to the vector bundle U 2 , yields the honest vector bundle U 1 . The functoriality of KU 0 is described via the pullback of vector bundles, i.e., if f : X → Y is a continuous map and U is a complex vector bundle on Y , one can form the vector bundle, (f * U ) x = U f (x) . We hence obtain a homomorphism of abelian groups, f * : KU 0 (Y ) → KU 0 (X).
It turns out that KU 0 (X) = π 0 (Map(X, C 0 )), and so one can extend KU to a cohomology theory by setting KU −q (X) = π 0 (Map(X, C q )), see discussion in Sec. II B. While the definition based on maps to the classifying space C q exhibits the additive structure on K-theory through loop concatenation in C q ΩC q−1 , the cohomology theory KU also admits a multiplicative structure. The tensor product of vector bundles, given by x endows KU 0 with a multiplicative structure, and this in turn extend to all other degrees, turning KU into a multiplicative generalized cohomology theory. Namely, we have a multiplication, KU p (X) ⊗ KU q (X) → KU p+q (X), for every space X, which is also super-commutative, i.e., uv = (−1) pq vu for u ∈ KU p (X) and v ∈ KU q (X). This multiplication endows KU * (X) def = q KU q (X) with a super-commutative graded ring structure. In the case where X = pt, the ring structure on KU * (pt) reveals a tight relationship between the different levels of K-theory. In fact, there is a class ξ ∈ KU −2 (pt) = KU 0 (S 2 ), which is invertible in the ring KU * (pt) and such that KU 2 (pt) = Z · ξ − . One summarizes this situation by the following shorthand notation, Here, the elements in the brackets represent generators of the ring KU * (pt) and |ξ| stands for the degree of ξ in the cohomology ring, e.g., if |u| = q then u ∈ KU q (pt). The complex Bott-periodicity is now just multiplication by ξ −1 ; see Table III.

b. Complex conjugation & real K-theory
The cohomology theory KU admits a canonical action of Z T 2 , through its compatible action on the C q -s. At degree q = 0, this action corresponds to the conjugation of complex vector bundles. Namely, a complex vector bundle U has a complex conjugateŪ , given byŪ x def = U x . On KU * (pt), this action is given bȳ Using the ring structure, we see that this formula is imposed by the simple fact thatξ = −ξ, along with the observation that U →Ū is a ring homomorphism: The action of Z T 2 through complex conjugation on complex vector bundles endows the cohomology theory KU with a Z T 2 -equivariant structure, i.e., turns it into a cohomology theory defined on topological spaces endowed with a Z T 2 -action. The resulting Z T 2 -equivariant cohomology theory, introduced by Atiyah in Ref. 141, is denoted by KR. Being Z T 2 -equivariant, the cohomology theory KR is naturally bi-graded ; we can form the (p, q)-cohomology, KR p,q (X), for a space X with a Z T 2action. This bi-grading is induced from the two natural based circles with Z T 2 -action: The one with the trivial action, denoted simply S 1 , and the one with the action by reflection along an axis, denotedS 1 . Namely, we have KR p,q (S d ∧S d ∧(X pt)) KR p−d,q−d (X), where X pt is the disjoint union of X and a point.
The bi-graded KR-theory of the point is slightly more complicated than its non-equivariant counterpart KU. Namely, KR * , * (pt) def = p,q KR p,q (pt) admits the following presentation, as a super-commutative graded ring [116,117], by generators and relations: where the (p, q)-degrees of the generators are given by see Refs. 142 and 143. These generators span the abelian groups summarized in Table III (cf. Table I) and the relations yield the multiplication rules summarized in Table IV. The group KR 0,0 (X) for a space X with a Z T 2 -action given by the involution, ι : X → X, can be interpreted as follows. It is the abelian group spanned by complex vector bundles U on X and endowed with an involutive identification, κ : U x ∼ → U ι(x) , which depends continuously on x ∈ X. The classes of complex vector bundles are subject to the relation [U ] + [U ] = [U ⊕ U ]. In particular, if ι acts trivially on X, we recover the group of real vector bundles on X, hence we have an identification of KR p,0 (X) with the real, eight-fold periodic, K-theory of X, usually denoted KO p (X).
In fact, via this relation, one is ought to think of KO as being the fixed points of KR by the Z T 2 -action. Similarly, we introduce a Z T 2 -equivariant cohomology theory KC, by assigning a Z T 2 -space its KU-cohomology, regardless of the Z T 2 -action. Then, we can view KU as the fixed points of KC. The bi-graded KC-cohomology of the point consist of infinitely many shifted copies of KU * (pt), namely, The above cohomology theories are related by some maps between them, reflecting operations on real and complex vector spaces. We have a complexification map, c : KR → KC, which forgets the identification of a vector bundle with its twisted complex conjugate. In the other direction, we have a (non-multiplicative) realification map, r : KC → KR, which takes a vector bundle U to the sum, U ⊕ ι * Ū , on which the map κ is given by (B25) Note, that these operations descend to the fixed points and provide maps KO KU, but we shall only consider the Z T 2 -equivariant versions in this paper. The effect of these maps on the generators is summarized in Table III. We shall explore the precise role and origin of these maps via the language of Z T 2 -spectra, introduced in the next section.

Spectra and equivariant spectra
a. The category of spectra To define a cohomology theory, such as KU, one have to provide an infinite delooping of a pointed space X 0 . Namely, to find a sequence of pointed spaces X i and homotopy equivalences, ΩX i X i−1 , which extrapolates the sequence X −i def = Ω i X 0 to all integers. From such a sequence, we get a generalized cohomology theory by taking the i-th cohomology of X to be π 0 (Map(X , X i )). A sequence of the form {X i } i∈Z as above is classically known as a spectrum, and in this case X 0 is an infinite loop space. To see that this construction provides a cohomology theory, one has to introduce an abelian group structure on π 0 (Map(X , X i )). This structure is induced from the identification, π 0 (Map(X , X i )) π 2 (Map(X , X i+2 )), via the canonical abelian group structure on π 2 of a space by concatenation of loops-of-loops. The above description of the addition in the cohomology theory KU is, however, unnecessarily complicated.
We have defined the additive group structure on KU 0 (X) via the direct sum operation on complex vector bundles. Thinking of this operation as a result of a loop concatenation in a delooping of the classifying space C 0 is a less straight-forward construction. One may think of the situation as the exact opposite. We can consider the existence of the addition of vector bundle as the reason, rather than the consequence, of a delooping to C 0 [144]. This line of thought leads to a model of spectra which is much more algebraic, as we shall now see.
As mentioned above, the notion of a spectrum is analogous to that of an abelian group. We can think of spectra as a version of abelian groups where one replaces the set of elements with a space of elements, i.e., a nice enough topological space, well defined up to homotopy equivalence. For every two spaces, X and Y , we can form the space of maps Map(X, Y ), allowing continuous families of maps, X → Y , depending on a continuous parameter. We denote the collection of spaces by Sps. Such a structure, in which maps between objects form a space, is called an ∞-category, see Ref. 145. We shall not get into the details of what that precisely means, but informally, this is a machinery designed to encode algebraic and geometric structures in which identities are satisfied only up to coherent homotopy. We shall soon see examples of this phenomenon.
It is now natural to ask what a commutative group in Sps should be. We start with the notion of a commutative monoid. Recall that a commutative monoid is a set, M , with a commutative and associative binary operation, +, which admits a zero element, 0. Similarly, one can define a commutative monoid in Sps, to be a space M endowed with a binary operation, +, which is associative and commutative up to coherent homotopy and with a zero object, 0, which is neutral to the addition up to coherent homotopy. Let us denote the collection (in fact, ∞-category) of such commutative monoids by CMon(Sps). To give a flavor of what that means, we replace, for example, the demand that the associativity relation (a+b)+c = a+(b+c) holds, with the structure of a specified homotopy between the left-hand side and the right-hand side of this equation, varying continuously in (a, b, c) ∈ M 3 . Similarly, we require a choice of a homotopy between the expressions, a + b and b + a, depending continuously on (a, b) ∈ M 2 , and a further hierarchy of higher homotopies connecting these basic ones.
A (usual) commutative monoid M is an abelian group, precisely when the map, M × M → M × M , given by is a bijection. Similarly, we say that a commutative monoid, M ∈ CMon(Sps), is group-like if the map, M × M → M × M , given by the same formula is a homotopy equivalence. It turns out that a group-like commutative monoid is the same as a spectrum with only non-negative homotopy groups. Namely, to specify a group-like commutative monoid, one has to specify a sequence of pointed spaces {X i } with chosen homotopy equivalences, ΩX i X i−1 , and such that the connectivity assumption, π i (X j ) = 0 for i < j, holds. This is a special case of a general principle called the May recognition principle [146], saying concisely that In other words, all group-like additions on a space are canonically given by concatenation of loops in a suitable delooping. Nevertheless, it is usually more natural to allow other types of group-like monoid structures, without explicitly identifying them with loop concatenation, as mentioned above. We shall be interested in spectra which have negative homotopy groups, such as KU. To achieve this, we note that if M ∈ CMon(Sps), then ΩM is automatically a group-like commutative monoid, and the loop concatenation and pointwise addition on ΩM coincide up to homotopy. Spectra are obtained from commutative monoids in Sps by formally inverting the loop operation, Objects of Spct then correspond to sequences {X i } without the connectivity assumption. The operation Ω −1 is denoted Σ, as it is related, but not to be confused with, the suspension operation on pointed spaces. If E is a spectrum, we can now define its homotopy groups π i (E) for all integer values i ∈ Z. If i ≥ 0, we take π i (E) to be the i-th homotopy of the commutative monoid corresponding to E. For negative homotopies, we set π −i (E) = π 0 (Σ i E).

b. Constructions in the category of spectra
The analogy, Spct ≈ Ab, suggests that constructions in abelian groups may be adapted to spectra. For example, an abelian group A has an underlying set of elements, obtained by forgetting the addition. Similarly, a spectrum E has an underlying space, E ∈ Sps. It has the property that π i (E) = π i (E) for i ≥ 0. For every two spectra, E and E , there is a mapping space, Map(E , E) ∈ Sps. In fact, similarly to the fact that Hom Ab (A, B) for abelian groups, A and B, has a canonical abelian group structure, there is a mapping spectrum, with an underlying space Map(E , E). The abelian group, Z, has as an analog the sphere spectrum, S. It is the free spectrum on a point, just like Z is the free abelian group on a point. More generally, if X is a set, we have the free abelian group, Similarly, there is a free spectrum on a space, S[X], characterized by the property, For historical reasons, this operation is usually denoted Σ ∞ , but we use S[−] to emphasize the analogy with the free abelian group construction. For two abelian groups, A and B, we can form their direct sum, A⊕B. There is a similar operation for spectra, compatible with homotopy groups. Moreover, we have . There is also a tensor product of abelian groups, A ⊗ B, classifying bilinear maps from A × B. Similarly, Spct has a symmetric monoidal structure, ⊗, which is uniquely characterized by the property that S[X × Y ] for X, Y ∈ Sps and is compatible with the gluing of spectra in the appropriate sense; see section 4.8.2 of Ref. 38. This operation is classically known as the smash product and usually denoted E ∧ E for spectra E and E . In particular, the tensor product of spectra distributes over the direct sum, Having a symmetric monoidal structure ⊗, we can define commutative rings and modules over them, as in section 4.5 of Ref. 38. Namely, a commutative ring in Spct is a spectrum R endowed with a multiplication map, R ⊗ R → R, and a unit, S → R, which are associative, commutative, and unital up to coherent homotopy. Similarly, a module over such a ring R is a spectrum M endowed with an action, a : R ⊗ M → M , which is associative up to coherent homotopy, and for which the unit of R acts as the identity up to coherent homotopy.
Spectra also give rise to cohomology theories. If E is a spectrum and X is a space, we can form the spectrum, Then, we obtain the cohomology by setting It can be shown that this definition give a cohomology theory for every spectrum E, hence relating spectra to cohomology theories, cf. Eq. (11). Similarly, one can turn a spectrum E into a homology theory by setting c. The K-theory spectrum & the monoid of vector spaces The cohomlogy theory KU, determined by the infinite loop space {C q }, has a multiplicative structure turning it into a multiplicative cohomology theory. Our aim now is to directly describe a construction of the spectrum KU as a commutative monoid and lift its multiplicative structure to Spct.
Let Vect C denote the collection of complex vector spaces. We can view Vect C as a space, whose points are complex vector spaces and the space of maps from U 1 to U 2 is the (possibly empty) space of invertible linear transformations from U 1 to U 2 .
The direct sum operation of vector spaces turns Vect C into a commutative monoid in Sps. Of course, it is not group-like, as one can not subtract vector spaces. To turn it into a group-like commutative monoid, one formally adds inverses to the elements of Vect C . This operation, known as group completion, is the homotopytheoretic analog of the way KU 0 (X) is obtained from the monoid of vector bundles on X by adding inverses to the elements. We denote the group completion of a commutative M by M grp . The spectrum Vect grp C is then a version of KU with only non-negative homotopy groups. To get KU, one then inverts the class, ξ ∈ π 2 (Vect grp C ). More precisely, the tensor product of vector spaces endows the group-like commutative monoid Vect grp C with a commutative ring structure in Spct and π 2 (Vect grp C ) Z is generated by a class ξ. Then, formally adding its inverse, we get KU = Vect grp C [ξ −1 ], which is a commutative ring spectrum lifting the cohomology theory KU * (X) to the spectral level; namely, Similarly to the spectra, we can define an ∞-category, Spct G , of spectra with a G-action. To do so, we mimic the line of thought presented in the previous section, but replacing spaces with G-spaces, i.e., spaces endowed with a G-action. Namely, let Sps G be the ∞-category whose objects are CW-complexes with action of the finite group G, and whose mapping spaces Map Sps G (X, Y ) consist of maps, f : X → Y , for which f (g(x)) = g(f (x)) for every g ∈ G and x ∈ X.
As for the non-equivariant version, we can define (group-like) G-commutative monoids. The reader is warned that these are not just commutative monoids in Sps G . Namely, to define such a monoid, one should specify summation maps not only for I-indexed families of elements of the monoid, whenever I is a finite set of indices. Rather, we should have such a map for every finite set I with, possibly non-trivial, G-action. For example, if G = Z 2 , then a G-monoid, E, should have not only a G-equivariant summation map, E × E → E, but also a trace map, into the Z 2 -fixed points, given informally by a → a + ι(a) for ι the generator of Z 2 . See Ref. 147 for further details.
To obtain the category Spct G , one inverts the formation of loops on G-commutative monoids: The ∞-category, Spct G , is endowed with a tensor product, ⊗, and moreover, for every pair, E, E ∈ Spct G , there is a spectrum of homomorphisms, Hom G (E, E ) ∈ Spct.
There is a 'free G-spectrum construction, S[−] : Sps G → Spct G , analogous to the non-equivariant version, which similarly turns disjoint unions to direct sums and products to tensor products.
The constructions of Sps G and Spct G for various groups, G, interact. Namely, if X is a G-space and H ⊆ G, we can regard X as an H-space by restriction of the G-action to H. We denote this operation by either Res G H (X) or X| H . Conversely, if X is an H-space we can turn it into a G-space in two different ways. On one hand, we can turn X into a G-space via X → G × H X, informally described as the quotient of G × X by the relation (gh, x) ∼ (g, h(x)), where G acts via the first coordinate. This construction satisfies Map G (G × H X, Y ) Map H (X, Y | H ). On the other hand, we can consider the construction, X → Map H (G, X), with G acting on the source. This construction satisfies Shifting our attention to spectra, the spectral analog of the functor, G × H X, is the functor, Ind G H : Spct H → Spct G . It satisfies as well as an adjunction relation, similar to the induction and restriction of representations.
The spectral analog of the functor, Map H (G, X), is more complicated, and called the Hopkins-Hill-Ravenel norm; see Ref. 148. It is informally defined by N G H (E) = G/H E for E ∈ Spct H , with the action of G given by permuting the factors and acting on each of them simultaneously. The norm is related to Map H (G, X) via The Hopkins-Hill-Ravenel norm interacts with the direct sum and tensor product of H-spectra in a way which we shall now describe. Namely, first of all, being itself an iterated tensor product, it commutes with the tensor product, so that The interaction with the direct sum operation is more complicated. If H ⊆ G is a subgroup, and E 1 , . . . , E m are H-spectra, we want to decompose N G H ( m i=1 E i ). To explicitly present this decomposition, more notation is needed. For a function, f : G/H → {1, . . . , m}, denote The set, G/H, decomposes into G f -orbits, and we denote the set of orbits by G f \G/H. We then have with an appropriate G-action.
If E ∈ Spct G , then we can define π n (E) = π 0 (Hom G (S, Ω n E)). More generally, if ρ : G → O(d) is a real linear representation of G, we can define a Gspectrum, S ρ , as follows. Denote by S ρ ∈ Sps G the G-space obtained from R d via one point compactification, on which G acts via ρ. We set S ρ to be the fiber of the map, S[S ρ ] → S, obtained from the constant map S ρ → pt by applying S[−]. In particular, since S ρ has a G-fixed point at the north pole of the one point compactification, we have The construction, ρ → S ρ , nicely interacts with the tensor product and the norm of G-spectra. For example, This allows us to define the RRep(G)-graded homotopy groups, In fact, this makes sense also when ρ is a virtual representation. The behavior of the Hopkins-Hill-Ravenel norm on representation spheres is also relatively simple. Namely, we have where Ind G H stands for the induction of real representations.
e. The G-equivariant K-theory spectrum Having the language of G-spectra set up, we can discuss the spectral versions of equivariant K-theory. We start with KR. The commutative monoid in spaces, TABLE XVI. All possible KR-module homomorphisms. Any particular spectra-homomorphism descends to a KR * , * (pt)-linear map, for which the images of all v ∈ KR p,q (pt) and u ∈ KC p,q (pt) in all bi-degrees (p, q) are determined by at most two elements, either v ∈ KR * , * (pt) or u , u ∈ KC * , * (pt). The last column contains the complexified KC * , * (pt)-linear maps.

Spectra-homomorphism
Elements KR-theory map KC-theory map Vect C , admits an involution, given by complex conjugation of complex vector spaces. This involution is obtained from a Z T 2 -equivariant commutative monoid structure on Vect C , for which the fixed points are Vect R and the trace map, Vect C → Vect R , is the realification map.
Applying an equivariant version of group completion, we get a Z T 2 -commutative ring in Spct Z T 2 . We get KR by inverting the classes β and µ. Since Rep(Z T 2 ) Z ⊕ Z, the homotopy groups of KR assemble to a bi-graded commutative ring, KR p,q (pt) = π −p,−q (KR); cf. Eq. (B49). Hence, KR is a commutative ring in Spct Z T 2 , lifting the Z T 2 -equivariant cohomology theory, X → KR * , * (X). A more complicated, yet similar, construction provides the G-equivariant K-theory, KR G . It is a G × Z T 2equivariant commutative ring spectrum obtained via Ggroup completion (and inverting certain classes) from a G-commutative monoid structure on Vect C , whose Hfixed points, for H ⊆ G, are the monoids in Sps consisting of H-representations. For H 0 ⊆ H 1 ⊆ G, the trace map associated with the surjection, G/H 0 → G/H 1 , for this G-commutative monoid, is given by induction of representations from H 0 to H 1 .
To get the analogous KC G , one applies the general induction and restriction machinery constructed in the previous section. Namely, In particular, the underlying spectrum of KR G is KU, while the underlying spectrum of KC G is KU ⊕ KU. The Z T 2 action for KR G corresponds to the conjugation action on KU, while for KC G it corresponds to the action, (u, u ) → (u ,ū), on KU ⊕ KU. The complexification map, c : KR G → KC G , then corresponds to a map, KU → KU⊕KU, given by u → (u,ū). Similarly, the realification map, r : KC G → KR G , corresponds to a summation map, KU⊕KU → KU, given by (u, u ) → u+ū . The reader might find these formulas similar to the realification and complexification maps on R and C. Indeed, the relation between KC and KR is a categorification of the relation between C and R, replacing real/complex numbers by real/complex vector spaces, addition by direct sum, and multiplication by tensor product.
We shall now return to the main reason we are interested in these spectral level constructions, that is, the classification of maps between KR-modules (see Ta analogous to the fact that C ⊗ R C C ⊕ C. While derived from fairly abstract considerations, it is not hard to translate those formulas into maps of cohomology theories. We consider them as constraints on the possible maps of the associated Z T 2 -equivariant cohomology theories in presence of a genuine KR-linear structure on the map, lifting the naïve π * , * (KR)-modules structure. Specifically, a map, KR → KR, of bi-degree (p, q) is given by multiplication by a class in π p,q (KR) KR −p,−q (pt). A map, KR → KC, of any bi-degree is of the form v → u c(v) where u ∈ π * , * (KC). A map, KC → KR, is of the form u → r(u u) for u ∈ π * , * (KC). Finally, a map, KC → KC, is of the form u → u u + u ū for u , u ∈ π * , * (KC). Any of these maps may be complexified by tensoring with KC. This produces π * , * (KC)linear maps by the identifications, The above is summarized in Table XVI. This is the precise meaning of our claim that the map ai in all bi-degrees depend on a small number of parameters: Being a map of KR-modules, it is specified by a class, or at most two classes, in either complex or real K-theory per matrix coefficient, for all bi-degrees (p, q) combined.

Signed permutation representations
Each space-group symmetry has a normal subgroup of translations corresponding to a Bravais lattice. Its reciprocal lattice, Λ, defines the geometry of the BZ torus, BZ = R 3 /Λ, with an action of the point-group, G. Sincē T 3 = S[BZ] plays an important role in the topological classification, we hereby describe the generic construction of T Λ for any G-lattice Λ.
One way to produce periodic lattices with G-action is by starting with a linear representation, ρ : G → O(d), and a basis, b 1 , . . . , b d , such that for every i = 1, . . . , d there is j = 1, . . . , d for which ρ(g)b i = ±b j . We call such a lattice, a signed permutation lattice for G, and ρ a signed permutation representation. If G acts on a lattice Λ ⊆ R d , one may form the torus, Let ρ : G → Aut(Λ) be a signed permutation representation, with basis, b 1 , . . . , b n , for which ρ(g)b i = ±b j and let Λ be the lattice generated by this basis. We wish to describe the G-spectrum, T Λ . The construction, Λ → T Λ , satisfies and so it essentially suffices to describe T Λ for indecomposable lattices Λ. In light of the previous discussion, we may assume that Λ is indecomposable, so that G permutes the lines spanned by the b i -s transitively. Let Λ 1 = Z · b 1 ⊆ R and G 1 be the the stabilizer of the line R · b 1 in R d . We have a G 1 -equivariant projection, P : T Λ → R/Z · b 1 , given by P (a 1 b 1 + . . . + a n b d ) = a 1 b 1 . By Eq. (B39), this projection determines a G-equivariant map, The map in Eq. (B58) is a G-equivariant homeomorphism and, in particular, a G-equivariant homotopy equivalence. Using Eq. (B42) and Eq. (B58), we see that To finish the analysis, let ρ 1 : G 1 → O(1) denote the representation of G 1 on the line R · b 1 . We have a direct sum decomposition, It follows that T Λ N G G1 (S G1 ⊕ S ρ1 G1 ). Combining Eq. (B45), Eq. (B48), and Eq. (B50), we can finally give the following description of T Λ for arbitrary signed permutation lattices. Each orbit of G on b 1 , . . . , b d may be represented by a set I k ⊆ {1, . . . , d} such that k=1 I k = {1, . . . , d}. Let G k be the stabilizer of I k and ρ k : G → O(|I k |) be the representation of G k on the linear span of {b i } i∈I k . We have the stable equivariant splitting, Hence, the study of the G-equivariant K-theory of the torus of a signed permutations representation reduces to that of representation spheres, which is covered by Refs. 62 and 123.

a. Quadratic representations & Pin structures
For a finite group G, the representations of G come into play in the computation of the G-equivariant Ktheory in two different, interacting ways. First, as seen in Eq. (B61), the G-spectrum T Λ decomposes as a sum of representation sphere spectra. Second, the G-equivariant K-theory spectrum itself encodes Grepresentations. Hence, we shall now review some of the features of the theory that come into play in both aspects and in the interaction between them.
Recall that we can think of a real representation of G as a map, ρ : G → GL(V ), for a real vector space V . We are mostly interested in representations which are orthogonal with respect to a symmetric bilinear form. Namely, let Q = Q ij be a symmetric bilinear form on V . We denote by O(Q) the group of linear maps, A : V → V , such that AQA T = Q, or, in terms of a basis, such that O(p, q). By a quadratic representation of G we mean a linear space V endowed with a non-degenerate bilinear form Q and a homomorphism, ρ : G → O(Q). Note, that O(Q) = O(−Q) and so a quadratic representation ρ on (V, Q) gives a quadratic representation ρ − on (V, −Q).
We recall the notion of a Pin-structure on a quadratic representation. For a quadratic form Q on a linear space V , let the Clifford algebra of Q, denoted Cl Q , be the free algebra generated by V , subject only to the relations, v 2 + Q(v, v) = 0, for v ∈ V . In particular, if Q = Id p ⊕ − Id q then Cl Q Cl p,q . We denote by Pin(Q) ⊆ Cl × Q the Pin group of Q, i.e., the subgroup of invertible elements in Cl Q of the form, a = v 1 · v 2 · · · v , for v i ∈ V . Recall that we have a canonical double cover, where the map, π : Pin(Q) → O(Q), takes v ∈ V to the reflection along the hyper-plane orthogonal to v. A Pinstructure on a quadratic representation, ρ : G → O(Q), is a homomorphic lift,ρ : G → Pin(Q) of ρ, i.e., such that the diagram, commutes. We warn that this notion depends heavily on Q. In particular, depending on whether Q is positive or negative definite, it coincides with the classical notion of Pin + and Pin − -structures respectively. Note, moreover, that a Pin-structure on a representation does not always exist. Even when it exists, a Pin-structure might not be unique; however, this will not affect our computations. If ρ admits no Pin-structure, one can always construct a double cover, Z 2 →G π − → G, such that the composition, G π → G ρ → O(Q), admits a Pin-structure. For example, if ρ is injective, we can takeG to be the preimage of G in Pin(Q) under π. In practice, we shall always assume that a Pin-structure exists on our representations, replacing G withG if necessary. This replacement have the effect of adding some irrelevant components to the K-theory that later will have to be identified and ignored; this technicality poses no practical troubles.
The 2-cocycle ω determines the double cover,Ĝ. Accordingly, for a G-space X we have a splitting, KRĜ(X) KR G (X) ⊕ KR ω G (X).
Hence, replacing G byĜ, we may reduce the computation of twisted K-theory to the computation of untwisted K-theory. For spinful fermions, all the formulas presented in Sec. II and Sec. III apply for KRĜ; the results in Table II are easily obtained by dropping the abelian groups stemming from the unphysical KR G (X) summand in Eq. (B66). This is explicitly demonstrated in Sec. IV B.
b. The Clifford group-algebra of a quadratic representation In Ref. 62, the G-equivariant K-theory of a representation sphere S ρ was described in terms of the Z 2 -graded algebra Cl Q [G], i.e., the algebra generated from G and Cl Q subject to the relation, gvg −1 = ρ(g)(v). To exploit this description, the authors had to decompose Cl Q [G] into a sum of Clifford algebras. the result becomes cleaner given Pin-structure on the representation ρ. Namely, for Z 2graded algebras, A and B, let A⊗B be their Z 2 -graded tensor product, i.e., the algebra generated from A and B subject to the relation, ab = (−1) |a||b| ba. Then, if ρ : G → O(Q) is a quadratic representation which admits a Pin structure, we have Note that, in particular, the algebra Cl Q [G] depends only on ρ . This "separation of variables" results from a change of coordinates [62,123]. Given a real representation ρ of G, one obtains a Z 2 -grading, ρ : G → Z 2 , on G, for which g ρ − − → det(ρ(g)), i.e., the parity of g ∈ G is sign of the determinant. Letρ : G → Pin(Q) be a pin structure on ρ. For every g ∈ G, set g def = gρ(g) −1 . (B68) We claim that these elements generate a copy of R ρ [G] in Cl Q [G] which super-commutes with Cl Q . To see this, first note that the conjugation action of Pin(Q) on itself, is given by ava −1 = (−1) |a| π(a)v for v ∈ V and a ∈ Pin(Q), see Eq. (B63). Hence, for every v ∈ V , we have Similarly, the elements g satisfy the same multiplication table as that of G: g 1 g 2 = g 1ρ (g 1 ) −1 g 2ρ (g 2 ) −1 = g 1ρ (g 1 ) −1 g 2ρ (g 2 ) −1 = g 1 g 2ρ (g 2 ) −1ρ (g 1 ) −1ρ (g 2 )ρ(g 2 ) −1 = g 1 g 2ρ (g 1 g 2 ) −1 = (g 1 g 2 ) .
These two identities prove that the generators, v ∈ V and g , together generate an algebra isomorphic to Cl Q⊗ R ρ [G].

c. Diagrammatic formulation
Let us make things more explicit by focusing on a 3D physical system. On one hand, for any G ⊂ O(3) we may always construct the double point-group,Ĝ ⊂ Pin − (3), that encodes the physical twist corresponding to the intrinsic spin-structure of spinful fermions. On the other hand, any subgroup, G k ⊆ G, which stabilizes a ddimensional sub-torus, acts on it via G k G k ⊂ O(d). It therefore has a different double cover,G k ⊂ Pin − (d), required for the separation of variables Eq. (B67). Nevertheless, one can always replace G k with its quadruplecover,Ĝ k , which admits both Pin-structures: Z 2 (B71) In particular, for any G k ⊆ G with geometric actions via O(3) and O(d), one can always explicitly construct (e.g., as a GAP4 [129] language algorithm) all the above groups and homomorphisms such that the diagram commutes and all sequences along straight lines are short (or long) exact sequences.

d. The equivariant K-theory of representation spheres
For a quadratic representation, ρ : G → O(Q), we associate a G × Z T 2 -equivariant spectrum S ρ , generalizing the construction of the representation sphere from Sec. B 3 d.
Let ρ : G → O(Q) be a quadratic representation of G. We can always decompose Q as a G-invariant direct sum Q = Q 1 ⊕ Q 2 such that Q 1 is positive definite and Q 2 is negative definite. We then set The uniqueness of the decomposition, Q = Q 1 ⊕ Q 2 , up to homotopy, shows that this association is well-defined.
Our aim now is to give a description of KR G (S ρ ) ∈ Mod KR (Spct Z T 2 ) for a quadratic representation ρ in elementary terms.
Assume from now on that ρ admits a Pin structure, ρ : G → Pin(Q). As we have seen in Sec. B 1 e, we have a decomposition of R ρ [G] into a sum of matrix algebras over Clifford algebras of the form If Q is of signature (p, q) then the above decomposition, together with the separation of variables [Eq. (B67)], implies that This determines a decomposition of the G-equivariant Ktheory spectrum: It thus provides a lift of Eq. (20) to the ∞-category of modules over KR. While the computation above describes KR G (S ρ ) completely as a KR-module, we shall need in our computation of the ai map, a slightly more canonical description of its homotopy groups. By doing it for all representations, ρ, together, it suffices to describe the 0-th homotopy group of KR G (S ρ ).
Let RRep Z2 (Cl Q [G]) be the Z 2 -graded representation ring of Cl Q [G], so that an element of RRep Z2 (Cl Q [G]) is a formal integral combination of Z 2 -graded representations. Given a Z 2 -graded representation, V = (V 0 , V 1 ), of Cl Q [G], we can construct a G-equivariant vector bundle, W , on S ρ , as follows: We take the constant Gbundle, V 0 , on the upper hemisphere of S ρ , and take the constant G-bundle, V 1 , on the lower hemisphere. We then glue these two G-bundles along the equator of S ρ in the following way. Every point on the equator of S ρ can be seen as a unit vector in the representation space of ρ, and hence as an odd element, e ∈ Cl Q . We identify V 0 and V 1 at e using the module structure of V over Cl Q , which in particular provides a linear isomorphism, e : V 0 ∼ → V 1 . By sending the class of [V ] ∈ RRep Z2 (Cl Q [G]) to the class [W ] − dim(W ) ∈ KR G (S ρ ) one obtains a map, RRep Z2 (Cl Q [G]) → π 0 ( KR G (S ρ )) π 0 (KR G (S ρ )), (B76) which is the G-analog of the Atiyah-Bott-Shapiro map [116]. In fact, the resulting map factors through the quotient of RRep Z2 (Cl Q [G]) by the image of the restriction map, and induces an isomorphism, Res : This isomorphism is functorial with respect to maps of representations. A morphism, f : ρ 0 → ρ 1 , of quadratic representations, with corresponding quadratic forms, Q 0 and Q 1 , induces a map, S ρ0 → S ρ1 , which in turn gives a map, f * : KR G (S ρ1 ) → KR G (S ρ0 ).
The map induced from f * on the 0-th homotopy group translates via Eq. (B77) to the restriction map, , along the algebra homomorphism, Cl Q0 [G] → Cl Q1 [G] induced from f .

Atomic insulators
As discussed in Sec. II A, besides the determination of the G-equivariant K-theory of the BZ torusT d , or equivalently, of the free G × Z T 2 -spectrumT d generated from it, we are also interested in the identification of the AI-bundles on the BZ. Namely, we wish to compute the map, ai p,q : AI p,q G → KR p,q (T 3 ).
We shall now explain how this map is computed uniformly in (p, q) for sign permutation representations.

a. Definition of AI-bundles
Let us first recall the definition of the AI-bundle associated with a representation of the little group G x for a Wyckoff position x. For such a data, We define the fundamental AI-bundle associated with x as follows. Let Λ denote the lattice of unit cell origins in the real (euclidean) space, E d . For every g ∈ G x , we have by definition that g(x) − x belongs to Λ. Hence, the function, is well defined on the BZ torus,T d . Consider the trivial one-dimensional vector bundle V on the BZ, with a nowhere vanishing global section, |ψ(k) . We endow V with the G-equivariant structure determined by the condition, g|ψ(k) = ν g x (k)|ψ(g(k)) .
that (x 1 , . . . , x d ) = ( k1 2π , . . . , k d 2π ). Since WP and HSM are reciprocal sets, we may now write ai as a "square matrix": ai p,q : x∈WP KR p,q Gx (pt) → KR p,q G (T d ) k∈HSM KR p,q G k (S ρ k ), (B86) where ρ k denotes the representation corresponding to the cell with center k. Let us denote by ai kx the (k, x) component of this matrix. Namely, ai kx is the composition, ai p,q kx : KR p,q Gx (pt) → AI p,q G ai − → KR p,q G (T d ) KR p,q G k (S ρ k ). (B87) The map, ai, in not diagonal, i.e., in general, ai kx = 0 even if k = x * . However, we shall now show that the quotient of KR p,q G (T ) by the image of of the map, ai, does not change if we replace the matrix by its diagonal. To obtain this, we shall use some dependencies between the different matrix entries of ai.
For every k ∈ HSM, letT k denote the coordinate torus containing k as its center. We have a G kequivariant projection map, P k :T d →T k . A key observation regarding the AI-bundles is that they are functorial with respect to sub-tori. Namely, for x ∈ WP, the position vector, g(x) − x, is orthogonal to the momenta in the cell with center x * . Hence, we have ν g x (k) = exp{i(g(x) − x) · k} = exp{i(g(x) − x) · P x (k)}, (B88) and consequently, v 1 x is in the image of the map, P * x * : KR p,q G (T x * ) → KR p,q G (T d ). More precisely, the pullback morphism, P * x * , takes the fundamental AI-bundle of the Wyckoff position x on the smaller torus,T x * , to the fundamental AI-bundle corresponding to the same Wyckoff position x for the BZ-torus,T d . If v ∈ KR p,q G k (T k ), we can decompose P * k (v) as a sum of components associated with the cells of the BZ torus. In fact, an element of the form P * k (v) has non-zero components only for the cells which are contained in the closure of the cell centered at k. It follows, that the matrix entries, ai kx , are non-zero only when x * ≥ k coordinate-wise, i.e., x * i ≥ k i for all i = 1, . . . , d.
We now analyse the case where x * ≥ k. The fundamental AI-bundle associated with x is constructed using the function ν g x (k), see Eq. (B80). Let x * 0 = k 0 denote a particular reciprocal pair. We wish to find the k 0component of the fundamental AI-bundle for all x 1 ≥ x 0 . Every x 1 ≥ x 0 can be written as x 1 = x 0 + x ⊥ , where x ⊥ is orthogonal to the torus with center k 0 . For every g ∈ G k0 and every k ∈T k0 , since k · x ⊥ = 0, one has ν g x1 (k) = exp{i(g(x 1 ) − x 1 ) · k} = exp{i(g(x 0 + x ⊥ ) − x 0 − x ⊥ ) · k} = exp{i(g(x 0 ) − x 0 ) · k + i(g(x ⊥ ) − x ⊥ ) · k} = exp{i(g(x 0 ) − x 0 ) · k} = ν g x0 (k), which is the function used to construct the fundamental AI-bundle of the smaller torus,T k0 . It follows that, up to appropriate inductions and restrictions, the entry, ai k0x1 , of the matrix ai is a multiple of the entry, ai k0x0 , i.e., of the corresponding diagonal entry. This readily implies that the upper triangular matrix, ai kx , can be diagonalized using invertible column operations. Hence, we have an isomorphism of the cokernel of ai and that of its diagonalized version, x ai x * x .

c. The spectral lift of atomic insulators
We now show that the collection of abelian groups, AI p,q G , and the collection of maps, ai p,q , are all the (p, q)pieces of a single map of KR-modules. We can view KR G as a ring in Spct G×Z T 2 , and we shall construct a KR G -module, AI G , together with a map of modules, ai : AI G → KR G (T ), which specialize to the desired lifts to KR-modules by forgetting the G-action.
We start with the groups AI p,q G . Let WP G denote the set of points in E d /Λ with half-integral entries in reciprocal primitive lattice vector coordinates so that WP is the set of representatives for each orbit of a point in WP G under the G-action on WP G . We can view WP G as a discrete object of the ∞-category Sps G×Z T 2 by letting Z T 2 act trivially. This entices the definition, To see that π −p,−q (AI G ) = AI p,q G , with respect to the previous definition of the AI groups, we note that S[WP G ] x∈WP Ind G Gx (S). Hence, by tensoring with KR G , we get Ind G Gx (KR G ).
(B91) Taking bi-graded homotopy groups we find, π −p,−q (AI G ) x∈WP π −p,−q (Ind G Gx (KR Gx )) x∈WP KR p,q Gx (pt). (B92) Hence, AI G is a lift of the bi-graded abelian group of AI-bundles. We now turn to the maps ai p,q . To define a lift of them to a map of modules over KR G , we need to specify a map, ai : KR G ⊗ S[WP G ] → KR G (T d ). The left-hand side is the free KR G -module on the finite G × Z T 2 -set, WP G . Hence, such a map is determined uniquely by a class in KR 0,0 G (WP G ×T d ) x∈WP KR 0,0 Gx (pt). Consider the function, (x, g, k) → ν g x (k), defined on WP G × G × T d . We can associate with it, as in the construction of the fundamental AI-bundles [Eq. (B81)], a G-equivariant vector bundle on WP G ×T d . This bundle yields the fundamental AI-bundles by restricting it to WP, which is the set of orbits of G on WP G . This provides the desired class in KR 0,0 G (WP G ×T d ).