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Equivalence classes of gapped Hamiltonians compatible with given symmetryconstraints, such as those underlying topological insulators, can be defined inmany ways. For the non-chiral classes modelled by vector bundles over Brillouintori, physically relevant equivalences include isomorphism, homotopy, and$K$-theory, which are inequivalent but closely related. We discuss an importantsubtlety which arises in the chiral Class AIII systems, where the windingnumber invariant is shown to be relative rather than absolute as is usuallyassumed. These issues are then analyzed and reconciled in the language of$K$-theory.

Topological phases: isomorphism, homotopy and $K$-theory