Online talk

We prove that the Madsen-Tillmann spectrum, appearing in the Galatius--Randal-Williams approach to moduli spaces of manifolds, splits into the sum of spectra  Σ^(-2n) MO<n+1> + Σ^(∞-2n) RP∞ _{2n} after Postnikov truncation. 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 the second homology groups of the diffeomorphism groups of the 2n-dimensional manifolds W_{g,1} for n >15 and g >6. This is joint work with Andrew Senger.