Topological quantum field theories and homotopy cobordisms

Abstract: We construct a category $\mathrm{HomCob}$ whose objects are {\it homotopically 1-finitely generated} topological spaces, and whose morphisms are {\it cofibrant cospans}. Given a manifold submanifold pair $(M,A)$, we prove that there exists functors into $\mathrm{HomCob}$ from the full subgroupoid of the mapping class groupoid $\mathrm{MCG}{M}^{A}$, and from the full subgroupoid of the motion groupoid $\mathrm{Mot}{M}^{A}$, whose objects are homotopically 1-finitely generated. We also construct a family of functors $\mathsf{Z}_G\colon \mathrm{HomCob}\to \mathbf{Vect}$, one for each finite group $G$. These generalise topological quantum field theories previously constructed by Yetter, and an untwisted version of Dijkgraaf-Witten. Given a space $X$, we prove that $\mathsf{Z}_G(X)$ can be expressed as the $\mathbb{C}$-vector space with basis natural transformation classes of maps ${\pi(X,X_0)\to G} $ for some finite representative set of points $X_0\subset X$, demonstrating that $\mathsf{Z}_G$ is explicitly calculable.