Speaker: Jonathan Sejr Pedersen – University of Toronto
Abstract: We prove that the Madsen-Tillmann spectrum $$MT\theta_n := Th(− \theta^{*}_ {n} \gamma_{2n}\longrightarrow BO(2n)\langle n\rangle)$$ splits into the sum of spectra $\Sigma^{−2n}MO⟨n⟩\oplus \Sigma^{\infty−2n}\mathbb{R}P^{\infty}_{2n}$ after Postnikov trunctation $\tau_{\leq l}$ for $l=\frac{n}{2}−c$. This is achieved by showing the connecting homomorphism $\tau_{\leq l}MO\langle n\rangle \longrightarrow \tau_{\leq l}\Sigma^{\infty+1}\mathbb{R}P^{\infty}_{2n}$ is nullhomotopic in this range by applying Adams spectral sequence arguments.
We discuss a number of applications, most prominently the computation of $H_2(B\mathrm{Diff}(W^{2n}_g,D^{2n});\mathbb{Z})$ which is connected to moduli spaces of high dimensional manifolds. This is joint work with Andy Senger.