Topological Quantum Field Theories and Homotopy Cobordisms

Abstract

AbstractWe construct a category $${\textrm{HomCob}}$$ 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 $${\textrm{HomCob}}$$ HomCob from the full subgroupoid of the mapping class groupoid $$\textrm{MCG}_{M}^{A}$$ MCG M A , and from the full subgroupoid of the motion groupoid $$\textrm{Mot}_{M}^{A}$$ Mot M A , whose objects are homotopically 1-finitely generated. We also construct a family of functors $${\textsf{Z}}_G:{\textrm{HomCob}}\rightarrow {\textbf{Vect}}$$ 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 $${\textsf{Z}}_G(X)$$ Z G ( X ) can be expressed as the $${\mathbb {C}}$$ C -vector space with basis natural transformation classes of maps from $$\pi (X,X_0)$$ π ( X , X 0 ) to G for some finite representative set of points $$X_0\subset X$$ X 0 ⊂ X , demonstrating that $${\textsf{Z}}_G$$ Z G is explicitly calculable.