Nontrivial homotopy and tunneling by topological instantons

Tunneling by topological instantons is described as a consequence of nontrivial homotopy among field histories and not of barrier penetration. A derivation of the Yang-Mills θ vacua, with finite-action (weak) boundary conditions, is given from this perspective which clarifies certain weaknesses of the barrier-penetration approach. The treatment of nontrivial homotopy in field-theory path integrals is discussed with special attention to the roles of finite action, compactification, continuity of paths, and the justification of the use of Euclidean instantons in a Minkowski-time path integral.