Adams spectral sequences for non-vector-bundle Thom spectra

TL;DR

This paper extends the use of Adams spectral sequences for computing twisted R-homology of spaces by removing the vector bundle twist assumption, enabling broader applications including supergravity anomaly analysis.

Contribution

It demonstrates that the vector bundle twist assumption is unnecessary by utilizing Baker-Lazarev's Adams spectral sequence for R-modules and computes new E2-pages for various twists.

Findings

Removed the vector bundle twist assumption in Adams spectral sequence computations.

Computed E2-pages for a broad class of twists of spectra like ku, ko, tmf, MTSpin^c, MTSpin, and MTString.

Applied the results to example computations related to supergravity anomaly cancellation.

Abstract

When $R$ is one of the spectra $\mathit{ku}$, $\mathit{ko}$, $\mathit{tmf}$, $\mathit{MTSpin}^c$, $\mathit{MTSpin}$, or $\mathit{MTString}$, there is a standard approach to computing twisted $R$-homology groups of a space $X$ with the Adams spectral sequence, by using a change-of-rings isomorphism to simplify the $E_2$-page. This approach requires the assumption that the twist comes from a vector bundle, i.e. the twist map $X\to B\mathrm{GL}_1(R)$ factors through $B\mathrm{O}$. We show this assumption is unnecessary by working with Baker-Lazarev's Adams spectral sequence of $R$-modules and computing its $E_2$-page for a large class of twists of these spectra. We then work through two example computations motivated by anomaly cancellation for supergravity theories.