Homotopy quantum field theory

Contents......Page 7
Introduction......Page 11
I.1 Basic definitions......Page 15
I.2 Cohomological HQFTs and transfer......Page 22
I.3 Aspherical targets......Page 25
I.4 Hermitian and unitary HQFTs......Page 30
I.5 Proof of Lemmas 1.3.1–1.3.3......Page 31
II.1 G-algebras......Page 37
II.2 Inner products and Frobenius G-algebras......Page 38
II.3 Crossed Frobenius G-algebras......Page 39
II.4 Transfer......Page 42
II.5 Semisimple crossed G-algebras......Page 45
II.6 Semisimple crossed Frobenius G-algebras......Page 50
II.7 Hermitian G-algebras......Page 52
III.1 The underlying G-algebra......Page 54
III.2 Computation for cohomological HQFTs......Page 63
III.3 Equivalence of categories......Page 66
III.4 Proof of Theorem 3.1......Page 68
III.5 Hermitian two-dimensional HQFTs......Page 77
IV.1 Frobenius G-algebras re-examined......Page 80
IV.2 Biangular G-algebras......Page 84
IV.3 Lattice HQFTs......Page 88
IV.4 Skeletons of surfaces......Page 95
IV.5 Hermitian biangular G-algebras......Page 99
V.1 Enumeration problem for homomorphisms......Page 102
V.2 Linear representations of Gamma and cohomology......Page 106
V.3 Projective representations of Gamma......Page 110
V.4 Properties of kappa_rho and zeta_rho......Page 115
V.5 Equivalence of two approaches......Page 121
V.6 A generalization and a proof of Theorem 1.2.1......Page 124
V.7 A homological obstruction to lifting......Page 129
V.8 Applications of Theorem 1.2.1......Page 135
V.9 Further applications of Theorem 1.2.1......Page 140
VI.1 G-categories......Page 144
VI.2 Crossed, braided, and ribbon G-categories......Page 147
VI.3 Colored G-tangles and their invariants......Page 154
VI.4 Colored G-graphs and their invariants......Page 163
VI.5 Trace, dimension, and algebra of colors......Page 168
VII.1 Modular crossed G-categories......Page 172
VII.2 Invariants of 3-dimensional G-manifolds......Page 177
VII.3 Homotopy modular functor......Page 184
VII.4 Two-dimensional HQFT......Page 190
VII.5 Three-dimensional HQFT......Page 193
VIII.1 Hopf G-coalgebras......Page 200
VIII.2 Canonical extensions......Page 207
VIII.3 Transfer of categories......Page 210
VIII.4 Quasi-abelian cohomology of groups......Page 213
VIII.5 Remarks on group-algebras......Page 214
Appendix 1.  Relative HQFTs......Page 219
Appendix 2.  State sum invariants of 3-dimensional G-manifolds......Page 227
Appendix 3.  Recent work on HQFTs......Page 231
Appendix 4.  Open problems......Page 233
5.1 Braided crossed G-categories......Page 235
5.2 The G-fixed category of a braided crossed G-category......Page 237
5.3 From braided categories containing Rep G to braided   G-crossed categories......Page 239
5.4 Classification and coherence for braided crossed         G-categories......Page 243
5.5 Braided crossed G-categories as crossed products......Page 244
5.6 Remarks on applications in conformal field theory......Page 247
6.1 Hopf G-coalgebras......Page 250
6.2 Quasitriangular Hopf G-coalgebras......Page 255
6.3 The twisted double construction......Page 259
7.1 Kuperberg-type invariants......Page 267
7.2 Hennings–Kauffman–Radford-type invariants......Page 272
Bibliography......Page 277
Index......Page 287