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ALGEBRAIC TOPOLOGY STEFAN FRIEDL Contents References 4 Initial remarks 7 1. Topological spaces 8 1.1. The definition of a topological space 8 1.2. Constructions of more topological spaces 16 1.3. Further examples of topological spaces 20 1.4. The two notions of connected topological spaces 23 1.5. Local properties 26 1.6. Graphs and topological realizations of graphs 28 1.7. The basis of a topology 31 1.8. Manifolds 33 1.9. The classification of 1-dimensional manifolds 36 1.10. Orientations of manifolds 40 2. Differential topology 43 2.1. The Tubular Neighborhood Theorem 43 2.2. The connected sum operation 47 2.3. Knots and their complements 48 3. How can we show that two topological spaces are not homeomorphic? 53 4. The fundamental group 57 4.1. Homotopy classes of paths 57 4.2. The fundamental group of a pointed topological space 63 5. Categories and functors 70 5.1. Definition and examples of categories 70 5.2. Functors 72 5.3. The fundamental group as functor 74 6. Fundamental groups and coverings 79 6.1. The cardinality of sets 79 6.2. Covering spaces 81 6.3. The lifting of paths 94 6.4. The lifting of homotopies 96 6.5. Group actions and fundamental groups 102 6.6. The fundamental group of the product of two topological spaces 108 1 2 STEFAN FRIEDL 6.7. Applications: the Fundamental Theorem of Algebra and the Borsuk-Ulam Theorem 111 7. Homotopy equivalent topological spaces 114 7.1. Homotopic maps 114 7.2. The fundamental groups of homotopy equivalent topological spaces 116 7.3. The wedge of two topological spaces 121 8. Basics of group theory 128 8.1. Free abelian groups and the finitely generated abelian groups 128 8.2. The free product of groups 134 8.3. An alternative definition of the free product of groups 141 9. The Seifert-van Kampen theorem I 144 9.1. The Seifert{van Kampen theorem I 144 9.2. Proof of the Seifert-van Kampen Theorem 9.2 151 9.3. More examples: surfaces and the connected sum of manifolds 156 10. Presentations of groups and amalgamated products 162 10.1. Basic definitions in group theory 162 10.2. Presentation of groups 163 10.3. The abelianization of a group 168 10.4. The amalgamated product of groups 172 11. The general Seifert-van Kampen Theorem 177 11.1. The formulation of the general Seifert-van Kampen Theorem 177 11.2. The fundamental groups of surfaces 179 11.3. Non-orientable surfaces 185 11.4. The classification of closed 2-dimensional (topological) manifolds 188 11.5. The classification of 2-dimensional (topological) manifolds with boundary 189 11.6. Retractions onto boundary components of 2-dimensional manifolds 194 12. Examples: knots and mapping tori 198 12.1. An excursion into knot theory (∗) 198 12.2. Mapping tori 203 13. Limits 211 13.1. Preordered and directed sets 211 13.2. The direct limit of a direct system 212 13.3. Gluing formula for fundamental groups and HNN-extensions (∗) 224 13.4. The inverse limit of an inverse system 229 13.5. The profinite completion of a group (∗) 237 14. Decision problems 239 15. The universal cover of topological spaces 242 15.1. Local properties of topological spaces 242 15.2. Lifting maps to coverings 243 15.3. Existence of covering spaces 248 16. Covering spaces and manifolds 261 16.1. Covering spaces of manifolds 261 ALGEBRAIC TOPOLOGY 3 16.2. The orientation cover of a non-orientable manifold 265 17. Complex manifolds 268 18. Hyperbolic geometry 274 18.1. Hyperbolic space 274 18.2. Angles in Riemannian manifolds 280 18.3. The distance metric of a Riemannian manifold 281 18.4. The hyperbolic distance function 285 18.5. Complete metric spaces 287 19. The universal cover of surfaces 289 19.1. Hyperbolic surfaces 289 19.2. More hyperbolic structures on the surfaces of genus g ≥ 2 (∗) 293 19.3. More examples of hyperbolic surfaces 295 19.4. The universal cover of surfaces 300 19.5. Proof of Theorem 19.9 I 301 19.6. Proof of Theorem 19.9 II 305 19.7. Picard's Theorem 308 20. The deck transformation group (∗) 312 21. Related constructions in algebraic geometry and Galois theory (∗) 323 21.1. The fundamental group of an algebraic variety (∗) 323 21.2. Galois theory (∗) 325 4 STEFAN FRIEDL References [A] S. I. Adyan, Algorithmic unsolvability of problems of recognition of certain properties of groups, Dokl. Akad. Nauk SSSR (N.S.) 103 (1955), 533{535. [AFW] M. Aschenbrenner, S. Friedl and H. Wilton, Decision problems for 3-manifolds and their fundamen- tal groups Baykur, R. Inanc (ed.) et al., Interactions between low dimensional topology and mapping class groups. Geometry and Topology Monographs 19 (2015), 201{236. [Ba] W. Ballmann, Lectures on K¨ahlermanifolds, ESI Lectures in Mathematics and Physics. Zrich: Euro- pean Mathematical Society Publishing House (2006). [BP] R. Benedetti and C. Petronio, Lectures on hyperbolic geometry, Universitext, Springer Verlag (1992) [Be] M. Berger, Geometry. I., Universitext, Springer Verlag (2009) [BS] A. Borel and J.-P. 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Grothendieck, Rev^etements´etaleset groupe fondamental, S´eminairede G´eom´etrieAlg´ebriquedu Bois-Marie, Lecture Notes in Mathematics 224 (1971). [Gr] I. A. Grushko, On the bases of a free product of groups, Matematicheskii Sbornik 8 (1940), 169{182. [Ha] A. Hatcher, Algebraic topology, Cambridge University Press (2002) https://www.math.cornell.edu/~hatcher/AT/AT.pdf [Ha2] A. Hatcher, Vector bundles and K-theory https://www.math.cornell.edu/~hatcher/VBKT/VB.pdf [Hi] M. Hirsch, Differential topology, Graduate Texts in Mathematics 33 (1976). [Hu] T. Hungerford, Algebra, Graduate Texts in Mathematics 73 (1980). [J1] K. J¨anich, Topology, Springer-Lehrbuch (2005). ALGEBRAIC TOPOLOGY 5 [J2] K. J¨anich, Funktionentheorie, 6. Auflage, Springer-Lehrbuch (2011). [Ke] M. Kervaire, Non-parallelizability of the n sphere for n > 7, Proc. N.A.S. 44 (1958), 280{283. [KM] M. Kervaire and J. Milnor, Groups of homotopy spheres: I, Ann. Math. 77 (1963), 504{537. [K] A. 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