Higher Chromatic Analogues of Twisted $K$-theory

TL;DR

This paper develops a new family of twisted $K(n)$-local theories analogous to twisted K-theory, introducing a framework for twisted $R_n$-theories with universal coefficient isomorphisms in the chromatic setting.

Contribution

It introduces higher chromatic analogues of twisted K-theory using homotopy fixed point spectra, expanding the scope of twisted cohomology theories.

Findings

Defined twisted $R_n$-theories for $K(n)$-local spaces with bundles.

Established a universal coefficient type isomorphism for these theories.

Connected the theories to classical twisted K-theory concepts.

Abstract

We introduce a family of twisted $K(n)$-local theories that behave analogous to twisted K-theory. Let $R_n= E_n^{hS\mathbb G_n}$, the homotopy fixed point spectrum under the action of the subgroup $S\mathbb G_n$ of the Morava stabilizer group where $S\mathbb G_n$ is the kernel of the determinant homomorphism $\text{det}:\mathbb G_n\to \mathbb Z_p^\times$. We show that for a $K(n)$-local space $X$ with a $L_{K(n)}K(\mathbb Z_p, n+1)$-bundle $P\to X$, the $P$-twisted $R_n$-theory of $X$ is defined. We show that analogous to twisted K-theory, a universal coefficient type isomorphism holds for these theories.