Topological Quantum Field Theories and Homotopy Cobordisms

We construct a category \({\textrm{HomCob}}\) whose objects are homotopically 1-finitely generated topological spaces, and whose morphisms are cofibrant cospans. Given a manifold submanifold pair (M, A), we prove that there exists functors into \({\textrm{HomCob}}\) from the full subgroupoid of the mapping class groupoid \(\textrm{MCG}_{M}^{A}\), and from the full subgroupoid of the motion groupoid \(\textrm{Mot}_{M}^{A}\), whose objects are homotopically 1-finitely generated. We also construct a family of functors \({\textsf{Z}}_G:{\textrm{HomCob}}\rightarrow {\textbf{Vect}}\), 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 \({\textsf{Z}}_G(X)\) can be expressed as the \({\mathbb {C}}\)-vector space with basis natural transformation classes of maps from \(\pi (X,X_0)\) to G for some finite representative set of points \(X_0\subset X\), demonstrating that \({\textsf{Z}}_G\) is explicitly calculable.