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C O NTENTS ...................................... .................... 3 PREFACE ............................ ..................................... 7 1. INTRODUCTION ........................................... ....... 11 1.1. Why symplectic and contact? ............................... 11 1.2. Results concerning Stein surfaces ........................... 13 1.3. Some contact results ...................................... . 21 2. TOPOLOGICAL SURGERIES ......................................... 25 2.1. Surgeries and handlebodies .................................. 25 2.2. Dehn surgery ................................................ 31 2.3. Kirby calculus ............................................... 38 3. SYMPLECTIC 4-MANIFOLDS ........................................ 49 3.1. Generalities about symplectic manifolds ................... 49 3.2. Moser's method and neighborhood theorems ............... 55 3.3. Appendix: The complex classification scheme for symplectic 4-m anifolds ............................................. ..... 58 4. CONTACT 3-MANIFOLDS ........................................... 63 4.1. Generalities on contact 3-manifolds ........................ 63 4.2. Legendrian knots ............................................ 72 4.3. Tight versus overtwisted structures ......................... 76 5. CONVEX SURFACES IN CONTACT 3-MANIFOLDS .................... 85 5.1. Convex surfaces and dividing sets ......................... 85 5.2. Contact structures and Heegaard decompositions ............ 96 6. Spinc STRUCTURES ON 3- AND 4-MANIFOLDS ...................... 99 6.1. Generalities on spin and spinc structures .................... 99 6.2. Spin0 structures and oriented 2-plane fields .................. 102 6.3. Spine structures and almost-complex structures .............. 105 7. SYMPLECTIC SURGERY ............................................ 111 7.1. Symplectic cut-and-paste ............... .............. 111 7.2. W einstein handles .......................... ................ 115 7.3. Another handle attachment .................................119 8. STEIN MANIFOLDS ................................................ 121 8.1. Recollections and definitions ................ ............ 121 8.2. Handle attachment to Stein manifolds ...................... 125 8.3. Stein neighborhoods of surfaces ............................ 127 9. OPEN BOOKS AND CONTACT STRUCTURES ..................... 131 9.1. Open book decompositions of 3-manifolds .................. 131 9.2. Compatible contact structures ........................... 138 9.3. Branched covers and contact structures .................... 150 10. LEFSCHETZ FIBRATIONS ON 4-MANIFOLDS ........................ 155 10.1. Lefschetz pencils and fibrations ............................. 155 10.2. Lefschetz fibrations on Stein domains ..................... 162 10.3. Som e applications .......................................... 173 11. CONTACT DEHN SURGERY ....................................... 179 11.1. Contact structures on S1 x D2 .......................... 179 11.2. Contact Dehn surgery .................... .............. 185 11.3. Invariants of contact structures given by surgery diagrams .. 191 12. FILLINGS OF CONTACT 3-MANIFOLDS ............................ 201 12.1. Fillings ................................. .... ............. . 201 12.2. Nonfillable contact 3-manifolds .......................... 206 12.3. Topology of Stein fillings ................................ 215 13. APPENDIX: SEIBERG-WITTEN INVARIANTS ...................... 223 13.1. Seiberg-Witten invariants of closed 4-manifolds ........... 223 13.2. Seiberg-Witten invariants of 4-manifolds with contact boun- dary ......................................................... 229 13.3. The adjunction inequality ................................. 231 14. APPENDIX: HEEGAARD FLOER THEORY .......................... 235 14.1. Topological preliminaries ................................... 235 14.2. Heegaard Floer theory for 3- and 4-manifolds ............. 239 14.3. Surgery triangles ................ ........................ 244 14.4. Contact Ozsvdth-Szab6 invariants ......................... 249 15. APPENDIX: MAPPING CLASS GROUPS ............................ 255 15.1. Short introduction ............. ...................... . 255 15.2. Mapping class groups and geometric structures ............. 263 15.3. Som e proofs ................. ........ ..................... 265 BIBLIOGRAPHY .................................................... 269 IN DEX ........................................ ...................... 279