A stable splitting of factorisation homology of generalised surfaces

For a manifold 𝑊 and an 𝐸𝑑-algebra 𝐴, the factorisation homology ∫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
Dif f 𝜕 (𝑊), and for the generalised surfaces 𝑊${}_{g,1}$ ∶=(#${}^g$𝑆${}^{𝑛}$×𝑆${}^{𝑛}$) ⧵̊D${}^{2𝑛}$, we have stabilisation maps among the quotients ∫${}_{𝑊g,1}$ 𝐴 ⫽ Diff${}_{𝜕}$ (𝑊${}_{g,1}$ which increase the genus g. In the case where a highly-connected tangential structure 𝜃 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 MT𝜃, and addresses the question of homological stability.