This book provides a unified combinatorial realization of the categroies of (closed, oriented) 3-manifolds, combed 3-manifolds, framed 3-manifolds and spin 3-manifolds. In all four cases the objects of the realization are finite enhanced graphs, and only finitely many local moves have to be taken into account. These realizations are based on the notion of branched standard spine, introduced in the book as a combination of the notion of branched surface with that of standard spine. The book is intended for readers interested in low-dimensional topology, and some familiarity with the basics is assumed. A list of questions, some of which concerning relations with the theory of quantum invariants, is enclosed.
Author(s): R. Benedetti, Carlo Petronio
Series: Lecture Notes in Mathematics
Edition: 1
Publisher: Springer
Year: 1997
Language: English
Pages: 141
Cover......Page 1
Series: Lecture Notes in Mathematics 1653......Page 2
Branched Standard Spines of 3-manifolds......Page 4
Copyright - ISBN: 3540626271......Page 5
Acknowledgements......Page 6
Contents......Page 8
1.1 Combinatorial realizations of topological categories......Page 10
1.2 Branched standard spines and an outline of the construction......Page 12
1.4 Statements of representation theore......Page 14
1.5 Existing literature and outline of contents......Page 19
2.1 Encoding 3-manifolds by o-graphs......Page 22
2.2 Reconstruction of the boundary......Page 26
2.3 Surgery presentation of a mirrored manifold and ideal triangulations......Page 29
3.1 Branchings on standard spines......Page 32
3.2 Normal o-graphs......Page 35
3.3 Bicoloration of the boundary......Page 37
3.4 Examples and existence results......Page 41
3.5 Matveev-Piergallini move on branched spines......Page 46
4.1 Oriented branchings and flows......Page 49
4.2 Extending the flow to a closed manifold......Page 54
4.3 Flow-preserving calculus: definitions and statements......Page 56
4.4 Branched simple spines......Page 59
4.5 Restoring the standard setting......Page 64
4.6 The MP-move which changes the flow......Page 69
5.1 Simple vs. standard branched spines......Page 73
5.2 The combed calculus......Page 78
6.1 Comparison of vector fields up to homotopy......Page 82
6.2 Pontrjagin moves for vector fields, and complete classification......Page 85
6.3 Combinatorial realization of closed manifolds......Page 90
7.1 The Euler cochain......Page 94
7.2 Framings of closed manifolds......Page 96
7.3 The framing calculus......Page 100
7.4 Spin structures on closed manifolds......Page 103
7.5 The spin calculus......Page 104
8.1 More on spin structures......Page 107
8.2 A review of recoupling theory and Reshetikhin-Turaev-Witten invariants......Page 108
8.3 Turaev-Viro invariants......Page 110
8.4 An alternative computation of TV invariants......Page 113
9.1 Internal questions......Page 117
9.2 Questions on invariants......Page 119
9.3 Questions on geometric structures......Page 125
10.1 Homology cohomology and duality......Page 130
10.2 More homological invariants......Page 132
10.3 Evenly framed knots in a spin manifold......Page 134
Bibliography......Page 136
Index......Page 140
