Topological Quantum Field Theories and Homotopy Cobordisms

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 MCGMA MCG M A , and from the full subgroupoid of the motion groupoid MotMA 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 π (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.