2001/05/02 by G. Rodrigues, Gracino Rodrigues, Rodrigues, G.
Mathematics · #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA) #math.AT #math.CT #math.QA
paper · pdf · doi:10.48550/arxiv.math/0105018
32 pages. Uses AMS-latex and xy-pic. Revised version: minor changes, correction of some typos, conclusions shortened. Proposition 1.3 corrected. To be published in Journal of Knot theory and its Ramifications
openalex publication_date 2001/05/02 · arxiv created 2002/08/14 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We define Homotopy quantum field theories (HQFT) as Topological quantum field theories (TQFT) for manifolds endowed with extra structure in the form of a map into some background space X. We also build the category of homotopy cobordisms HCobord(n,X) such that an HQFT is a functor from this category into a category of linear spaces. We then derive some very general properties of HCobord(n,X), including the fact that it only depends on the (n+1)-homotopy type of X. We also prove that an HQFT with target space X and in dimension n+1 implies the existence of geometrical structures in X; in particular, flat gerbes make their appearance. We give a complete characterization of HCobord(n,X) for n=1 (or the 1+1 case) and X the Eilenberg-Maclane space K(G,2). In the final section we derive state sum models for these HQFT's.