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This note is about the topology of the path space of linear Fredholm operators on a real Hilbert space. Fitzpatrick and Pejsachowicz introduced the parity of such a path, based on the Leray–Schauder degree of a path of parametrices. Here an alternative analytic approach is presented which reduces the parity to the    Z  2   ‐valued spectral flow of an associated path of chiral skew‐adjoints. Furthermore, the related notion of    Z  2   ‐index of a Fredholm pair of chiral complex structures is introduced and connected to the parity of a suitable path. Several non‐trivial examples are provided. One of them concerns topological insulators, another an application to the bifurcation of a non‐linear partial differential equation.

Parity as Z 2 ‐valued spectral flow