Topological Quantum Field Theories and Homotopy Cobordisms

Abstract We construct a category HomCob 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 HomCob HomCob from the full subgroupoid of the mapping class groupoid MCG₌^A MCG M A, and from the full subgroupoid of the motion groupoid Mot₌^A Mot M A, whose objects are homotopically 1-finitely generated. We also construct a family of functors ZG: HomCob Vect Z G: HomCob → 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 ZG (X) Z G (X) can be expressed as the {C} C -vector space with basis natural transformation classes of maps from (X, X₀) π (X, X 0) to G for some finite representative set of points X₀ X X 0 ⊂ X, demonstrating that ZG Z G is explicitly calculable.