Computations in higher twisted $K$-theory

TL;DR

This paper explores higher twisted K-theory, extending classical twisted K-theory to include all homotopy-theoretic twists, providing new geometric representations and computational tools like spectral sequences.

Contribution

It reformulates higher twisted K-theory from a topological perspective and develops methods for explicit geometric representatives and computations using spectral sequences.

Findings

Explicit geometric representatives of higher twists are constructed.

Spectral sequences are developed for computational purposes.

Computations are performed for various topological spaces.

Abstract

Higher twisted $K$-theory is an extension of twisted $K$-theory introduced by Ulrich Pennig which captures all of the homotopy-theoretic twists of topological $K$-theory in a geometric way. We give an overview of his formulation and key results, and reformulate the definition from a topological perspective. We then investigate ways of producing explicit geometric representatives of the higher twists of $K$-theory viewed as cohomology classes in special cases using the clutching construction and when the class is decomposable. Atiyah-Hirzebruch and Serre spectral sequences are developed and information on their differentials is obtained, and these along with a Mayer-Vietoris sequence in higher twisted $K$-theory are applied in order to perform computations for a variety of spaces.