Orientation flow for skew-adjoint Fredholm operators with odd-dimensional kernel

Abstract

The orientation flow of paths of real skew-adjoint Fredholm operators with invertible endpoints was studied by Carey, Phillips and Schulz-Baldes. For paths of real skew-adjoint Fredholm operators with odd-dimensional kernel the orientation flow is defined with respect to a real one-dimensional reference projection. It is homotopy invariant and fulfills a concatenation property. When applied to closed paths it is independent of the reference projection and provides an isomorphism of the fundamental group of the space of real skew-adjoint Fredholm operators with odd-dimensional kernel to Z2. As an example the orientation flow of the magnetic flux inserted in a half-sided Kitaev chain is studied.