Hakenness and B 1

1. Review of Thurston’s hyperbolization theorem and quasi-Fuchsian representations 2. Review of Thurston’s Surgery Theorem 3. Review of Freedman’s surfaces and Cooper-Long-Reid extension of Freedman’s work 4. Proof of the Surgery Theorem 5. A fundamental trihotomy for π1injective surfaces 6. Proof of the double coset theorem 7. Quasi-Fuchsian knots and analytic proof of Freedman-Cooper-Long theorem 8. Freedman’s surfaces give boundary groups, analysis à la Kerckhoff-Thurston 9 . Digression to Arithmetic Topology: Freedman’s surface and generalized cyclotomic units 10. Jaco’s theorem and its corollaries 11. p-cycles 12. Homology domination 13. Tight knots 14. Appendix: The language of arithmetic topology 15. Appendix: Hyperbolization of Heegard splitted manifolds and bounded width in the mapping class group 16. Appendix: Euler class and free generation 17. Appendix: Homology of ramified coverings