Geometric construction of classes in van Daele's K-theory

TL;DR

This paper provides explicit geometric generators for real K-theory of spheres, linking them to Hamiltonians in topological insulators, and extends bivariant K-theory to the real case for concrete computations.

Contribution

It introduces a geometric construction of K-theory classes for real spheres and extends bivariant K-theory to the real setting, enabling explicit calculations.

Findings

Explicit generators for real K-theory of spheres are constructed.

Pullbacks of generators produce Hamiltonians used in topological insulator models.

K-theory classes are computed geometrically using extended bivariant K-theory.

Abstract

We describe explicit generators for the "real" K-theory of "real" spheres in van Daele's picture. Pulling these generators back along suitable maps from tori to spheres produces a family of Hamiltonians used in the physics literature on topological insulators. We compute their K-theory classes geometrically, based on wrong-way functoriality of K-theory and the geometric version of bivariant K-theory, which we extend to the "real" case.