Jonathan Sejr Pedersen

arXiv2025

Splitting the Madsen-Tillmann Spectra $MTθ_n$

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 $g \geq 7$.

math.AT↗

arXiv2023

On Generators and Relations of the Rational Cohomology of Hilbert Schemes

We consider for $d\geq 1$ the graded commutative $\mathbb{Q}$-algebra $\mathcal{A}(d):=H^*(\operatorname{Hilb}^d(\mathbb{C}^2);\mathbb{Q})$, which is also connected to the study of generalised Hurwitz spaces by work of the first author. These Hurwitz spaces are in turn related to the moduli spaces of Riemann surfaces with boundary. We determine two distinct, minimal sets of $\lfloor d/2\rfloor$ multiplicative generators of $\mathcal{A}(d)$. Additionally, we prove when the lowest degree generating relations occur. For small values of $d$ we also determine a minimal set of generating relations, which leads to several conjectures about the necessary generating relations for $\mathcal{A}(d)$.

math.AC↗