Splitting the Madsen-Tillmann Spectra $MTθ_n$
Jonathan Sejr Pedersen, Andrew Senger
TL;DR
The paper proves that the Madsen–Tillmann spectrum MTθ_n splits after Postnikov truncation in a range ℓ = ⌊n/2⌋ − 6, into a sum of the shifted Thom spectrum Σ^{-2n}MO⟨n+1⟩ and a shifted stunted projective spectrum Σ^{∞-2n}RP^{∞}_{2n}. The key method is an Adams-filtration analysis that shows the attaching map in the MTθ_n cofiber sequence becomes nullhomotopic in that range, using an auxiliary spectrum Y built from stable cohomotopy Euler classes and 2–primary control, complemented by rational and fracture-square arguments for odd primes. As applications, the authors compute H_2 of the classifying space of diffeomorphism groups BDiff(W^{2n}_g,D^{2n}) for large n and g, expressing it in terms of the homotopy groups of MO⟨n+1⟩ and RP^{∞}_{2n}. The work connects to GRW’s abelianization results and the broader context of hermitian K-theory, providing a framework to extend higher–homotopy information of MTθ_n via surgical splitting. Overall, the paper advances the understanding of the stable homotopy type of MTθ_n and its consequences for high-dimensional manifold bundles.
Abstract
We prove that the Madsen-Tillmann spectrum $MTθ_n$ splits into the sum of spectra $Σ^{-2n}MO\langle n+1 \rangle \oplus Σ^{\infty-2n}\mathbb{R} P^\infty_{2n}$ after Postnikov trunctation $τ_{\leq \ell}$ 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 $g \geq 7$.