Localization formulae in odd K-theory

TL;DR

This paper explores the geometric and topological structures of classifying spaces for odd K-theory using Grassmannians of lagrangian subspaces, and relates these to spectral flow and cohomology generators.

Contribution

It introduces a geometric framework for odd K-theory via Grassmannians and Schubert varieties, linking spectral flow to cohomology classes.

Findings

Classifies $K^{-1}$ using Grassmannians of lagrangian subspaces.

Describes cohomology generators via incidence relations and Schubert varieties.

Identifies spectral flow as the first cohomology generator.

Abstract

We describe a class of real Banach manifolds, which classify $K^{-1}$. These manifolds are Grassmannians of (hermitian) lagrangian subspaces in a complex Hilbert space. Certain finite codimensional real subvarieties described by incidence relations define geometric representatives for the generators of the cohomology rings of these classifying spaces. Any family of self-adjoint, Fredholm operators parametrized by a closed manifold comes with a map to one of these spaces. We use these Schubert varieties to describe the Poincare duals of the pull-backs to the parameter space of the cohomology ring generators. The class corresponding to the first generator is the spectral flow.