Splitting Madsen-Tillmann spectra I. Twisted transfer maps

TL;DR

This paper explores properties of twisted transfer maps in algebraic topology, demonstrating new splitting results for Madsen-Tillmann spectra and implications for characteristic classes, especially at the prime 2.

Contribution

It introduces new splitting theorems for Madsen-Tillmann spectra and analyzes their consequences for characteristic classes and cohomology.

Findings

$BSO(2n+1)_+$ splits off $MTO(2n)$

$MTO(2n)$ is homotopy equivalent to $BO(2n)_+$ after localization away from 2

Identifies algebraically independent classes in mod 2 cohomology of $\

Abstract

We record various properties of twisted Becker-Gottlieb transfer maps and study their multiplicative properties analogous to Becker-Gottlieb transfer. We show these twisted transfer maps factorise through Becker-Schultz-Mann-Miller-Miller transfer; some of these might be well known. We apply this to show that $BSO(2n+1)_+$ splits off $MTO(2n)$, which after localisation away from $2$, refines to a homotopy equivalence $MTO(2n)\simeq BO(2n)_+$ as well as $MTO(2n+1)\simeq *$ for all $n\geqslant0$. This reduces the study of $MTO(n)$ to the $2$-localised case. At the prime $2$ our splitting allows to identify some algebraically independent classes in mod $2$ cohomology of $\Omega^\infty MTO(2n)$. We also show that $BG_+$ splits off $MTK$ for some pairs $(G,K)$ at appropriate set of primes $p$, and investigate the consequences for characteristic classes, including algebraic independence and non-divisibility of some universally defined characteristic classes, generalizing results of Ebert and Randal-Williams.