Preface ....................................................... vii
1 Preliminaries ................................................ 1
1.1 Sets .................................................... 1
1.2 Manifolds ............................................... 1
1.3 Curves .................................................. 2
1.4 Transversality .......................................... 3
1.5 Regular deformations .................................... 4
1.6 Complexes ............................................... 6
2 Filling Dehn surfaces ........................................ 9
2.1 Dehn surfaces in 3-manifolds ............................ 9
2.2 Filling Dehn surfaces .................................. 11
2.3 Notation ............................................... 13
2.4 Surgery on Dehn surfaces. Montesinos Theorem ........... 15
2.4.1 Type 0 arcs ..................................... 16
2.4.2 Type 1 arcs ..................................... 17
2.4.3 Type 2 arcs ..................................... 18
2.4.4 Surgeries ....................................... 19
2.4.5 Spiral piping ................................... 21
3 Johansson diagrams .......................................... 25
3.1 Diagrams associated to Dehn surfaces ................... 25
3.2 Abstract diagrams on surfaces .......................... 26
3.3 The Johansson Theorem .................................. 30
3.4 Filling diagrams ....................................... 41
4 Fundamental group of a Dehn sphere .......................... 51
4.1 Coverings of Dehn spheres .............................. 51
4.2 The diagram group ...................................... 53
4.3 Coverings and representations .......................... 54
4.4 Applications ........................................... 58
4.5 The fundamental group of a Dehn g-torus ................ 60
5 Filling homotopies .......................................... 63
5.1 Filling homotopies ..................................... 63
5.2 Bad Haken moves ........................................ 68
5.3 "Not so bad" Haken moves ............................... 70
5.4 Diagram moves .......................................... 72
5.5 Duplication ............................................ 77
5.6 Amendola's moves ....................................... 82
6 Proof of Theorem 5.8 ........................................ 85
6.1 Pushing disks .......................................... 85
6.2 Shellings. Smooth triangulations ...................... 103
6.2.1 Shellings ...................................... 103
6.2.2 Smooth triangulations .......................... 106
6.2.3 Shellings of 2-disks ........................... 108
6.3 Complex ƒ-moves ....................................... 111
6.3.1 Finger move 3/2 ................................ 111
6.3.2 Singular saddles ............................... 113
6.3.3 Pushing disks along a 2-cells .................. 115
6.3.4 Pushing disks along 3-cells .................... 118
6.3.5 Inflating double points ........................ 122
6.3.6 Passing through spiral pipings ................. 125
6.4 Inflating triangulations .............................. 130
6.4.1 Inflating T .................................... 130
6.4.2 Naming the regions of ΣT ....................... 134
6.4.3 ΣT fills M ..................................... 137
6.4.4 Inflating filling immersions ................... 139
6.5 Filling pairs ......................................... 144
6.6 Simultaneous growings ................................. 148
6.7 Proof of Theorem 5.8 .................................. 148
7 The triple point spectrum .................................. 153
7.1 The Shima's spheres ................................... 153
7.2 Some examples of filling Dehn surfaces ................ 158
7.2.1 A filling Dehn sphere in  2 × 1 ............... 158
7.2.2 A filling Dehn torus in 2 × 1 ................ 158
7.2.3 A filling Dehn sphere in L(3,1) ................ 160
7.3 The number of triple points as a measure of
complexity: Montesinos complexity ..................... 162
7.4 The triple point spectrum ............................. 169
7.5 Surface-complexity .................................... 171
8 Knots, knots and some open questions ....................... 173
8.1 2-Knots: lifting filling Dehn surfaces ................ 173
8.2 1-Knots ............................................... 175
8.3 Open problems ......................................... 177
8.3.1 Filling Dehn surfaces and filling Dehn
spheres ........................................ 177
8.3.2 Filling homotopies. Moves ...................... 177
8.3.3 Montesinos complexity. Triple point spectrum ... 178
8.3.4 Knots .......................................... 180
Appendix A Proof of Key Lemma 2 .............................. 183
Appendix В Proof of Lemma 6.46 ............................... 237
Appendix С Proof of Proposition 6.57 ......................... 251
Bibliography .................................................. 267
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