Twisted K-theory, Real $\mathcal{A}$-bundles and Grothendieck-Witt groups

TL;DR

This paper develops a unified framework for various twisted topological K-theories, highlighting how Grothendieck-Witt groups serve as key examples linked to real algebraic vector bundles.

Contribution

It introduces a general approach connecting twisted K-theory variants with Grothendieck-Witt groups and real algebraic bundles, expanding the theoretical understanding.

Findings

Grothendieck-Witt groups exemplify twisted K-theory variants

Unified framework for multiple twisted K-theories

Connection established between algebraic vector bundles and Grothendieck-Witt groups

Abstract

We introduce a general framework to unify several variants of twisted topological $K$-theory. We focus on the role of finite dimensional real simple algebras with a product-preserving involution, showing that Grothendieck-Witt groups provide interesting examples of twisted $K$-theory. These groups are linked with the classification of algebraic vector bundles on real algebraic varieties.