A stable splitting of factorisation homology of generalised surfaces

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 $\mathrm{Diff}{}_{\partial }\left(W\right)$, 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 ${\int }_{W_{g,1}}A⫽\mathrm{Diff}{}_{\partial }\left(W_{g,1}\right)$ which increase the genus $g$. In the case where a highly-connected tangential structure $\theta$ is taken into account, this article describes the stable homology of these quotients in terms of the iterated bar construction $B^{2n}A$ and a tangential Thom spectrum $\mathrm{MT}\theta$, and addresses the question of homological stability.