A stable splitting of factorisation homology of generalised surfaces
Florian Kranhold
TL;DR
This work develops a general, homotopy-theoretic description of stable factorisation homology for generalized, θ-framed surfaces. By organizing decorated embeddings into genus-graded operads and employing pushforward (O ⊗^L_P A) techniques, the author isolates a universal MT θ–type summand and an E_∞-algebra–encoded component through a sequence of operadic reductions and group-completion theorems. The main theorem provides a concrete loop-space decomposition for Ω B W_{*,1}^θ[A], featuring an A_∞-action of ΩL and a homotopy quotient of a shifted iterated bar construction B^{2n} UA, fibered over MT θ; this yields stable homology isomorphisms in a range and connects to classical stability results (Harer, Galatius–RW, BBPTY). The framework unifies and extends Bonatto’s decorated configuration spaces to arbitrary d and tangential structures, enabling a robust description of stabilized moduli spaces and their homology in terms of MT θ and iterated bar constructions.
Abstract
For a manifold $W$ and an $E_d$-algebra $A$, the factorisation homology $\int_W A$ can be seen as a generalisation of the classical configuration space of labelled particles in $W$. It carries an action by the diffeomorphism group $\mathrm{Diff}_\partial(W)$, and for the generalised surfaces $W_{g,1}:=(\#^g S^n\times S^n)\setminus\mathring D{}^{2n}$, we have stabilisation maps among the quotients $\int_{W_{g,1}} A\,/\!/\,\mathrm{Diff}_\partial(W_{g,1})$ which increase the genus $g$. In the case where a highly-connected tangential structure $θ$ is taken into account, we describe its stable homology in terms of the iterated bar construction $\mathrm{B}^{2n}A$ and a tangential Thom spectrum $\mathrm{MT}θ$. We also consider the question of homological stability.