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

We prove that the Madsen-Tillmann spectrum $MT{\theta }_n$ splits into the sum of spectra ${\mathrm{\Sigma }}^{-2n}MO⟨n+1⟩\oplus {\mathrm{\Sigma }}^{\mathrm{\infty }-2n}\mathbb{R}{P}_{2n}^{\mathrm{\infty }}$ after Postnikov trunctation ${\tau }_{\le \mathrm{\ell }}$ for $\mathrm{\ell }=\lfloor \frac{n}{2}\rfloor -6$. 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 $g\ge 7$.