We prove that the Madsen-Tillmann spectrum $MT\theta_n$ splits into the sum of spectra $\Sigma^{ -2n}MO\langle n+1 \rangle \oplus \Sigma^{ \infty-2n}\mathbb{ R} P^\infty_{ 2n}$ after Postnikov trunctation $\tau_{ \leq \ell}$ 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 $H_2(B\operatorname{ Diff}(W^{ 2n}_{ g},D^{ 2n});\mathbb{ Z})$ up to extensions for $n \geq 16$ and $g \geq 7$.
Categories: math.AT