Florian Kranhold: A stable splitting of factorisation homology of generalised surfaces

Abstract.

For a manifold \(W\) and an \(E_d\)-algebra \(A\), the factorisation homology of \(W\) with coefficients in A admits an action by the diffeomorphism group of \(W\) and we consider its homotopy quotient \(W[A]\). For \(W_{g,1}=D^{2n} \# (\#^g S^n×S^n)\), the collection of all \(W_{g,1}[A]\) is a monoid by taking boundary-connected sums. We discuss its homological stability and describe its group-completion in terms of a tangential Thom spectrum and the iterated bar construction of \(A\). We do so by identifying the above collection with an algebra over the generalised surface operad, establishing a splitting result for such algebras, and studying the free infinite loop space over a given (framed) \(E_d\)-algebra.