Projective Spaces and Splitting of Madsen-Tillmann Spectra

Takuji Kashiwabara, Hadi Zare

Published 2014-07-27, updated 2015-01-31Version 2

We show that $BSO(2n+1)_+$ splits off $MTO(2n)$, which after localisation away from $2$, refines to a homotopy equivalence $MTO(2n)\simeq BO(2n)_+\simeq BSO(2n+1)_+\simeq BSp(n)_+$ as well as $MTO(2n+1)\simeq *$ for all $n\geqslant0$. This reduces the problem of studying $MTO(n)$, with $n\geqslant0$, to the study of its $2$-local version; at the prime $2$ our splitting allows to identify some algebraically independent torsion classes in homology of $\Omega^\infty MTO(2n)$. The classes arising from $BSO(2n+1)_+$ have been known to exist in the case of $n=1$ due to work of Randal-Williams. We also show that $BSpin(2n+1)_+$ splits off $MTPin_{-}(2n)$ which leads to existence of some algebraically independent classes in homology of $\Omega^\infty MTPin_{-}(2n)$, and that $BSU(n+1)_+$ splits off $MTU(n)$ when localised at $p$ with $p\nmid(n-1)(n+1)$, hence locating some torsion classes in homology $\Omega^\infty MTU(n)$. As a corollary, $S^0$ splits off $MTO(2n)$, $MTU(n)$ and $MTSp(n)$ at certain primes; we have sharpened the conditions using Madsen-Tillmann-Weiss map. As an application, we find a family of algebraically independent elements among universally defined characteristic classes discussed by Randal-Williams. We also provide a nonexistence result of short exact sequences of Hopf algebras for an infinite family of Madsen-Tillmann spectra. For $n=1$, localised at the prime $2$, $MTO(2)$ splits as $BSO(3)_+\vee\Sigma^{-2}D(2)$, where the other summand is obtained using Steinberg idempotents which will be discussed in a subsequent work.