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june 2018 Splitting Madsen-Tillmann spectra I. Twisted transfer maps
Takuji Kashiwabara, Hadi Zare
Bull. Belg. Math. Soc. Simon Stevin 25(2): 263-304 (june 2018). DOI: 10.36045/bbms/1530065013

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 factor 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 $2$-localized case. At the prime $2$ our splitting allows us 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.

Citation

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Takuji Kashiwabara. Hadi Zare. "Splitting Madsen-Tillmann spectra I. Twisted transfer maps." Bull. Belg. Math. Soc. Simon Stevin 25 (2) 263 - 304, june 2018. https://doi.org/10.36045/bbms/1530065013

Information

Published: june 2018
First available in Project Euclid: 27 June 2018

zbMATH: 1431.55010
MathSciNet: MR3819126
Digital Object Identifier: 10.36045/bbms/1530065013

Subjects:
Primary: 55P42 , 55P47 , 55R12 , 57T25

Rights: Copyright © 2018 The Belgian Mathematical Society

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Vol.25 • No. 2 • june 2018
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