Turaev Vladimir. Homotopy Quantum Field Theory

European Mathematical Society Publishing House, 2010. — 290 p.In this monograph we apply the idea of a TQFT to maps from manifolds to topological spaces. This leads us to a notion of a (d+1)-dimensional homotopy quantum field theory (HQFT) which may be described as a TQFT for closed oriented d-dimensional manifolds and compact oriented (d+1)-dimensional cobordisms endowed with maps to a given space X . Such an HQFT yields numerical homotopy invariants of maps from closed oriented (d+1)-dimensional manifolds to X . A TQFT may be interpreted in this language as an HQFT with target space consisting of one point. The general notion of a (d+1)-dimensional HQFT was introduced in 1999 in my unpublished preprint [Tu3] and independently by M. Brightwell and P. Turner [BT1] for d=1 and simply connected target spaces.Generalities on HQFTs
Basic definitions
Cohomological HQFTs and transfer
Aspherical targets
Hermitian and unitary HQFTs
Proof of Lemmas 1.3.1–1.3.3
Group-algebras
G-algebras
Inner products and Frobenius G-algebras
Crossed Frobenius G-algebras
Transfer
Semisimple crossed G-algebras
Semisimple crossed Frobenius G-algebras
Hermitian G-algebras
Two-dimensional HQFTs
The underlying G-algebra
Computation for cohomological HQFTs
Equivalence of categories
Proof of Theorem 3.1
Hermitian two-dimensional HQFTs
Biangular algebras and lattice HQFTs
Frobenius G-algebras re-examined
Biangular G-algebras
Lattice HQFTs
Skeletons of surfaces
Hermitian biangular G-algebras
Enumeration problems in dimension two
Enumeration problem for homomorphisms
Linear representations of Г and cohomology
Projective representations of Г
Properties of Kp and /p
Equivalence of two approaches
A generalization and a proof of Theorem
A homological obstruction to lifting
Applications of Theorem 1.2.1
Further applications of Theorem 1.2.1
Crossed G -categories and invariants of links
G-categories
Crossed, braided, and ribbon G-categories
Colored G-tangles and their invariants
Colored G-graphs and their invariants
Trace, dimension, and algebra of colors
Modular G -categories and HQFTs
Modular crossed G-categories
Invariants of 3-dimensional G-manifolds
Homotopy modular functor
Two-dimensional HQFT
Three-dimensional HQFT
Miscellaneous algebra
Hopf G-coalgebras
Canonical extensions
Transfer of categories
Quasi-abelian cohomology of groups
Remarks on group-algebras
Appendix 1. Relative HQFTs
Appendix 2. State sum invariants of 3-dimensional G -manifolds
Appendix 3. Recent work on HQFTs
Appendix 4. Open problems
Appendix 5. On the structure of braided crossed G -categories (by Michael Müger)
Braided crossed G-categories
The G-fixed category of a braided crossed G-category
From braided categories containing Rep G to braided G-crossed categories
Classification and coherence for braided crossed G-categories
Braided crossed G-categories as crossed products
Remarks on applications in conformal field theory
Appendix 6. Algebraic properties of Hopf G -coalgebras (by Alexis Virelizier)
Hopf G-coalgebras
Quasitriangular Hopf G-coalgebras
The twisted double construction
Appendix 7. Invariants of 3-dimensional G -manifolds from Hopf coalgebras (by Alexis Virelizier)
Kuperberg-type invariants
Hennings–Kauffman–Radford-type invariants