Preface ....................................................... vii
1 4-manifold handlebodies ...................................... 1
1.1 Carving ................................................. 4
1.2 Sliding handles ......................................... 6
1.3 Canceling handles ....................................... 8
1.4 Carving ribbons ........................................ 10
1.5 Non-orientable handles ................................. 14
1.6 Algebraic topology ..................................... 16
2 Building low-dimensional manifolds .......................... 19
2.1 Plumbing ............................................... 21
2.2 Self plumbing .......................................... 22
2.3 Some useful diffeomorphisms ............................ 23
2.4 Examples ............................................... 24
2.5 Constructing diffeomorphisms by carving ................ 29
2.6 Shake shce knots ....................................... 32
2.7 Some classical invariants .............................. 33
3 Gluing 4-manifoIds along their boundaries ................... 37
3.1 Constructing - M ̮ ƒ N by the upside down method ........ 37
3.2 Constructing - M ̮ ƒ N and M(ƒ) by the cylinder method
(roping) ............................................... 39
3.3 Codimension zero surgery M → M' ........................ 42
4 Bundles ..................................................... 43
4.1 T4 = T2 × T2 ............................................ 43
4.2 Cacime surface ......................................... 45
4.3 General surface bundles over surfaces .................. 52
4.4 Circle bundles over 3-manifolds ........................ 52
4.5 3-manifold bundles over the circle ..................... 53
5 3-manifolds ................................................. 55
5.1 Dehn surgery ........................................... 56
5.2 Prom framed links to Heegaard diagrams ................. 57
5.3 Gluing knot complements ................................ 59
5.4 Carving 3-manifolds .................................... 61
5.5 Rohlin invariant ....................................... 62
6 Operations .................................................. 64
6.1 Gluck twisting ......................................... 64
6.2 Blowing down ribbons ................................... 67
6.3 Logarithmic transform .................................. 68
6.4 Luttinger surgery ...................................... 69
6.5 Knot surgery ........................................... 71
6.6 Rational blowdowns ..................................... 74
7 Lefschetz fibrations ........................................ 78
7.1 Elliptic surface E(n) .................................. 80
7.2 Dolgachev surfaces ..................................... 81
7.3 PALFs .................................................. 84
7.4 ALFs ................................................... 87
7.5 BLFs ................................................... 92
8 Symplectic manifolds ........................................ 95
8.1 Contact manifolds ...................................... 96
8.2 Stein manifolds ........................................ 98
8.3 Eliashberg's characterization of Stein ................. 99
8.4 Convex decomposition of 4-manifolds ................... 101
8.5 T4 = |BLF| ............................................ 103
8.6 Stein = |PALF| ........................................ 106
8.7 Imbedding Stein to symplectic via PALF ................ 107
8.8 Symplectic fillings ................................... 109
9 Exotic 4-manifolds ......................................... 112
9.1 Constructing small exotic manifolds ................... 112
9.2 Iterated 0-Whitehead doubles are non-slice ............ 116
9.3 A solution of a conjecture of Zeeman .................. 119
9.4 An exotic 4 .......................................... 119
9.5 An exotic non-orientable closed manifold .............. 120
10 Cork decomposition ......................................... 123
10.1 Corks ................................................. 125
10.2 Anticorks ............................................. 130
10.3 Knotting corks ........................................ 132
10.4 Plugs ................................................. 133
11 Covering spaces ............................................ 138
11.1 Handlebody of coverings ............................... 138
11.2 Handlebody of branched coverings ...................... 141
11.3 Branched covers along ribbon surfaces ................. 147
12 Complex surfaces ........................................... 150
12.1 Milnor fibers of isolated singularities ............... 150
12.2 Hypersurfaces that are branched covers of 2 ........ 153
12.3 Handlebody descriptions of Vd ......................... 155
12.4 ∑(a,b,c) .............................................. 158
13 Seiberg-Witten invariants .................................. 163
13.1 Representations ....................................... 164
13.2 Action of λ*(Х) on W* ................................. 166
13.3 Dirac operator ........................................ 171
13.4 A special calculation ................................. 173
13.5 Seiberg-Witten invariants ............................. 174
13.6 S-W when b2+(X) = l .................................... 181
13.7 Blowup formula ........................................ 182
13.8 S-W for torus surgeries ............................... 182
13.9 S-W for manifolds with T3 boundary .................... 183
13.10 S-W for logarithmic transforms ....................... 185
13.11 S-W for knot surgery ХK .............................. 186
13.12 S-W for S1 × YЗ ...................................... 189
13.13 Moduli space near the reducible solution ............. 191
13.14 Almost complex and symplectic structures ............. 193
13.15 Antiholomorphic quotients ............................ 200
13.16 S-W equations on × YЗ .............................. 201
13.17 Adjunction inequality ................................ 202
14 Some applications .......................................... 206
14.1 10/8 theorem .......................................... 206
14.2 Cappell-Shaneson homotopy spheres ..................... 210
14.3 Flexible contractible 4-manifolds ..................... 219
14.4 Some small closed exotic manifolds .................... 223
14.4.1 An exotic 2 # 32 ......................... 223
14.4.2 An exotic
2 # 22 .................................... 234
14.4.3 Fintushel-Stern reverse engineering ............ 243
References .................................................... 245
Index ......................................................... 261
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