Splitting the Madsen-Tillmann Spectra $MT{\theta }_n$

Abstract

We prove that the Madsen-Tillmann spectrum $MT{\theta }_n$ splits into the sum of spectra ${\Sigma }^{-2n}MO⟨n+1⟩\oplus {\Sigma }^{\infty -2n}\mathbb{R}{P}_{2n}^{\infty }$ after Postnikov trunctation ${\tau }_{\le \ell }$ for $\ell =\lfloor \frac{n}{2}\rfloor -6$. To accomplish this, we prove that the connecting map in a certain fiber sequence is nullhomotopic in this range by an Adams filtration argument. As an application, we compute $H_2(B\mathrm{Diff}(W_g^{2n},D^{2n});\mathbb{Z})$ up to extensions for $n\ge 16$ and $g\ge 7$.

13 pages, comments welcome!