algebraic topology i + ii - uni-regensburg.de...algebraic topology i + ii 5 33.2. homology groups of...

ALGEBRAIC TOPOLOGY I + II

STEFAN FRIEDL

Contents

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81. Topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.1. The definition of a topological space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.2. Constructions of more topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.3. Further examples of topological spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261.4. The two notions of connected topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 291.5. The (path-) components of a topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321.6. Local properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351.7. Graphs and topological realizations of graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371.8. The basis of a topology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401.9. Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421.10. The classification of 1-dimensional manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451.11. Orientations of manifolds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2. Differential topology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522.1. The Tubular Neighborhood Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522.2. The connected sum operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562.3. Knots and their complements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3. How can we show that two topological spaces are not homeomorphic? . . . . . . . . . 624. The fundamental group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.1. Homotopy classes of paths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.2. The fundamental group of a pointed topological space . . . . . . . . . . . . . . . . . . . . . 73

5. Categories and functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795.1. Definition and examples of categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795.2. Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.3. The fundamental group as functor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

6. Fundamental groups and coverings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 886.1. The cardinality of sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 886.2. Covering spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 906.3. The lifting of paths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1036.4. The lifting of homotopies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1056.5. Group actions and fundamental groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1116.6. The fundamental group of the product of two topological spaces . . . . . . . . . . . 117

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6.7. Applications: the Fundamental Theorem of Algebra and the Borsuk-UlamTheorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

7. Homotopy equivalent topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1237.1. Homotopic maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1237.2. The fundamental groups of homotopy equivalent topological spaces . . . . . . . . 1257.3. The wedge of two topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

8. Basics of group theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1378.1. Free abelian groups and finitely generated abelian groups . . . . . . . . . . . . . . . . . . 1378.2. The free product of groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1438.3. An alternative definition of the free product of groups . . . . . . . . . . . . . . . . . . . . . 150

9. The Seifert-van Kampen theorem I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1539.1. The Seifert–van Kampen theorem I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1539.2. Proof of the Seifert-van Kampen Theorem 9.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1609.3. More examples: surfaces and the connected sum of manifolds . . . . . . . . . . . . . . 165

10. Presentations of groups and amalgamated products . . . . . . . . . . . . . . . . . . . . . . . . . . . 17110.1. Basic definitions in group theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17110.2. Presentation of groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17210.3. The abelianization of a group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17710.4. The amalgamated product of groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

11. The general Seifert-van Kampen Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18611.1. The formulation of the general Seifert-van Kampen Theorem . . . . . . . . . . . . . 18611.2. The fundamental groups of surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18811.3. Non-orientable surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19411.4. The classification of closed 2-dimensional (topological) manifolds . . . . . . . . . 19711.5. The classification of 2-dimensional (topological) manifolds with boundary. 19811.6. Retractions onto boundary components of 2-dimensional manifolds . . . . . . . 203

12. Examples: knots and mapping tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20712.1. An excursion into knot theory (∗) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20712.2. Mapping tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

13. Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22013.1. Preordered and directed sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22013.2. The direct limit of a direct system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22113.3. Gluing formula for fundamental groups and HNN-extensions (∗) . . . . . . . . . . 23313.4. The inverse limit of an inverse system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23813.5. The profinite completion of a group (∗). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

14. Decision problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24815. The universal cover of topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25115.1. Local properties of topological spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25115.2. Lifting maps to coverings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25215.3. Existence of covering spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

16. Covering spaces and manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27016.1. Covering spaces of manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270

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16.2. The orientation cover of a non-orientable manifold. . . . . . . . . . . . . . . . . . . . . . . . 27417. Complex manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27718. Hyperbolic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28318.1. Hyperbolic space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28318.2. Angles in Riemannian manifolds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28918.3. The distance metric of a Riemannian manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . 29118.4. The hyperbolic distance function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29418.5. Complete metric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296

19. The universal cover of surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29819.1. Hyperbolic surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29819.2. More hyperbolic structures on the surfaces of genus g ≥ 2 (∗) . . . . . . . . . . . . . 30219.3. More examples of hyperbolic surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30419.4. The universal cover of surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30919.5. Proof of Theorem 19.9 I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31019.6. Proof of Theorem 19.9 II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31419.7. Picard’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317

20. The deck transformation group (∗) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32121. Related constructions in algebraic geometry and Galois theory (∗) . . . . . . . . . . . . 33221.1. The fundamental group of an algebraic variety (∗) . . . . . . . . . . . . . . . . . . . . . . . . 33221.2. Galois theory (∗) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334

22. CW-complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33622.1. Definition of finite-dimensional CW-complexes and examples . . . . . . . . . . . . . 33622.2. Two topologies on R∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34222.3. Infinite-dimensional CW-complexes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34522.4. Properties of CW-complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34622.5. The Homotopy Extension Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35522.6. Fundamental groups of CW-complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35722.7. The Cellular Approximation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36322.8. Proof of Proposition 22.20 (∗) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36722.9. Coverings of CW-complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37222.10. Spanning trees of graphs (∗) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376

23. Higher homotopy groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37923.1. Definition of the higher homotopy groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37923.2. Properties and calculations of the higher homotopy groups . . . . . . . . . . . . . . . 38623.3. Covering spaces and higher homotopy groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38823.4. Are there any higher homotopy groups that are non-trivial? . . . . . . . . . . . . . . 39123.5. The Poincaré Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394

24. The homology groups of a topological space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39824.1. Singular chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39824.2. Definition of the homology groups of a topological space . . . . . . . . . . . . . . . . . . 40124.3. First calculations of homology groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40624.4. Algebraic chain complexes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408

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24.5. The functoriality of homology groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41024.6. Direct products and direct sums (∗) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41124.7. The homology groups of a direct sum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412

25. Homology and homotopies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41425.1. Chain homotopies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41425.2. Homology and homotopic maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415

26. Long exact sequences and the homology of quotient spaces . . . . . . . . . . . . . . . . . . . 42226.1. Long exact sequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42226.2. The homology groups of spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42426.3. Basic homological algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42826.4. Relative homology groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43426.5. The Excision Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44026.6. The proof of the Excision Theorem 26.15: the idea . . . . . . . . . . . . . . . . . . . . . . . 44226.7. The proof of the Excision Theorem 26.15: the full details . . . . . . . . . . . . . . . . . 44326.8. Explicit generators of homology groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45626.9. Applications to topological manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462

27. The degree of a self-map of a sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46528. The Mayer–Vietoris sequence and its applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47328.1. Split exact sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47328.2. The Mayer–Vietoris sequence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47628.3. Applications of the Mayer–Vietoris sequence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47928.4. The Mayer–Vietoris Theorem for CW-complexes . . . . . . . . . . . . . . . . . . . . . . . . . 48128.5. The homology groups of the torus and the Klein bottle . . . . . . . . . . . . . . . . . . . 48228.6. The homology groups of a knot complement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48728.7. The homology groups of a mapping torus (∗) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489

29. Cellular homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49429.1. The homology groups of a nested sequence of topological spaces . . . . . . . . . . 49429.2. The definition of cellular homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49729.3. The relationship between cellular and singular homology . . . . . . . . . . . . . . . . . 50029.4. The cellular boundary maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50429.5. The homology groups of 2-dimensional manifolds . . . . . . . . . . . . . . . . . . . . . . . . . 50929.6. The local degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513

30. The Jordan Curve Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52331. Topological robotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53131.1. The matrix groups SO(3) and SU(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53131.2. Topological robotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534

32. The first homology group and the fundamental group. . . . . . . . . . . . . . . . . . . . . . . . . 53932.1. The Hurewicz homomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53932.2. Natural transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54532.3. The Hurewicz homomorphism in higher dimensions . . . . . . . . . . . . . . . . . . . . . . . 550

33. Simplicial complexes and homology groups of manifolds . . . . . . . . . . . . . . . . . . . . . . 55333.1. Simplicial complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553


33.2. Homology groups of manifolds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55533.3. Representing homology classes by manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56933.4. The degree of a map between oriented manifolds . . . . . . . . . . . . . . . . . . . . . . . . . 571

34. The Euler characteristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57534.1. The definition of the Euler characteristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57534.2. Properties of the Euler characteristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57734.3. Groups acting on spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58234.4. Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58334.5. The Lefschetz Fixed Point Theorem (∗) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584

35. Applications of the Euler characteristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58635.1. Building a leather football . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58635.2. Platonic solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58735.3. Homology spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59235.4. Planar graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594

36. Homology with coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59836.1. Chain complexes over rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59836.2. The tensor product of abelian groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59836.3. The tensor product of a chain complex with an abelian group . . . . . . . . . . . . 60236.4. Free resolutions and the G-torsion of an abelian group. . . . . . . . . . . . . . . . . . . . 60536.5. The Universal Coefficient Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61536.6. Applications to topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62036.7. The singular chain complex and the celluar chain complexes are chain

homotopic (∗) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62137. The homology groups of products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62437.1. The tensor product of chain complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62437.2. The product of CW-complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62537.3. The Eilenberg-Zilber Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62737.4. The Künneth-formula for chain complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 627

38. Applications of homology groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63338.1. Persistent homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63338.2. Division algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63438.3. The transfer map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63838.4. The Borsuk-Ulam Theorem and the Ham-Sandwich Theorem . . . . . . . . . . . . . 640

39. The homology groups of topological manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64639.1. The topology of topological manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64639.2. Orientations of topological manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65039.3. The fundamental class of closed topological manifolds . . . . . . . . . . . . . . . . . . . . 65439.4. The fundamental class of topological manifolds with boundary. . . . . . . . . . . . 66039.5. Chiral and amphichiral manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664

40. The cohomology groups of a topological space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66640.1. Dual cochain complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66640.2. The singular cohomology group of a topological space . . . . . . . . . . . . . . . . . . . . 667

6 STEFAN FRIEDL

40.3. Examples of cohomology groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67040.4. The cohomology groups of a direct system of topological spaces. . . . . . . . . . . 671

41. The Universal Coefficient Theorem for cohomology groups . . . . . . . . . . . . . . . . . . . . 67541.1. The Ext group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67541.2. The Universal Coefficient Theorem for Cohomology Groups . . . . . . . . . . . . . . 68041.3. The cohomology group of product topological spaces . . . . . . . . . . . . . . . . . . . . . 683

42. Cohomology with compact support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68642.1. Direct limits of directed systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68642.2. The definition of cohomology with compact support . . . . . . . . . . . . . . . . . . . . . . 687

43. The de Rham-cohomology and singular cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . 69343.1. Alternating forms and the wedge-product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69343.2. The de Rham-cohomology of a manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69543.3. The connection between the two cohomology theories . . . . . . . . . . . . . . . . . . . . . 69743.4. The wedge product of differential forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 700

44. The cup-product and the cap-product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70344.1. The definition of the cup-product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70344.2. Examples of the cup-product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70744.3. More on the cup-product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70944.4. The definition of the cap-product. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 711

45. Poincaré-duality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71545.1. The Poincaré Duality Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71545.2. A proof of a weaker version of the Poincaré Duality Theorem 45.1 . . . . . . . . 71845.3. The proof of the Poincaré Duality Theorem 45.1. . . . . . . . . . . . . . . . . . . . . . . . . . 722

46. The cup-product and algebraic intersection numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 72846.1. The cup-product of topological manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72846.2. Cup-products and degree-one maps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73546.3. Algebraic intersection numbers of submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . 73646.4. The cup-product and algebraic intersection numbers . . . . . . . . . . . . . . . . . . . . . . 74246.5. Representing homology classes by manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 747

47. The intersection form of topological manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74847.1. The definition of the intersection form of topological manifolds . . . . . . . . . . . 74847.2. The intersection form and the connected sum operation . . . . . . . . . . . . . . . . . . 75047.3. Algebra: Classification of non-singular forms over the real numbers . . . . . . . 75547.4. Algebra: Non-singular forms over the integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75847.5. Back to topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76047.6. The signature and finite covers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 761

48. Tubular Neighborhoods and Alexander duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76248.1. Tubular neighborhoods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76248.2. Alexander duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764

49. The linking form on rational homology spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76749.1. Rational homology spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76749.2. More on cohomology groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 767


49.3. The linking form on odd-dimensional rational homology spheres . . . . . . . . . . 770

8 STEFAN FRIEDL

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[Ni] L. Nicolaescu. An invitation to Morse theory, Universitext, Springer Verlag (2011)[Ol] Y. Ollivier. A January 2005 invitation to random groups, Ensaios Matemáticos 10. Rio de Janeiro:

Sociedade Brasileira de Matemática (2005).http://www.yann-ollivier.org/rech/publs/randomgroups.pdf

[OR] E. Outerelo and J. Ruiz. Mapping degree theory, Graduate Studies in Mathematics 108 (2009).[Pa1] R. Palais. Extending diffeomorphisms, Proc. Am. Math. Soc. 11 (1960), 274–277.[Pa2] R. Palais. The classification of real division algebras, Am. Math. Mon. 75 (1968), 366–368.[Pe1] G. Perelman. The entropy formula for the Ricci flow and its geometric applications,

arXiv:math.DG/0211159[Pe2] G. Perelman. Ricci flow with surgery on three-manifolds, arXiv:math.DG/0303109.[Pe3] G. Perelman. Finite extinction time for the solutions to the Ricci flow on certain three-manifolds,

arXiv:math.DG/0307245.[Qu] F. Quinn. Ends of maps. III: Dimensions 4 and 5, J. Differ. Geom. 17 (1982), 503–521.[Ra] M. Rabin. Recursive unsolvability of group theoretic problems, Ann. of Math. (2) 67 (1958), 172–194.

[Rd] T. Radó. Über den Begriff der Riemannschen Fläche, Acta Szeged 2 (1926), 101–121.[Rb] D. Robinson. A course in the theory of groups, 2nd ed, Graduate Texts in Mathematics 80, Springer

Verlag (1995).[Ro] D. Rolfsen. Knots and links, Mathematics Lecture Series. 7. Houston, TX: Publish or Perish. (1990).[RS] C. P. Rourke and B. J. Sanderson. Introduction to piecewise-linear topology, Ergebnisse der Mathe-

matik und ihrer Grenzgebiete 69 (1972).[Re] H. Rose. Linear Algebra. A pure mathematical approach, Birkhäuser Verlag, 2002.[Rs] J. Rosenberg. Algebraic K-theory and its applications, Graduate Texts in Mathematics 147, Springer

Verlag (1994).[Rt] J. Rotman. An introduction to algebraic topology, Graduate Texts in Mathematics 119, Springer Verlag

(1988)[Rt2] J. Rotman. An introduction to homological algebra, 2nd edition, Universitext, Springer Verlag (2009).[Ru] Y. Rudyak. On Thom spectra, orientability, and cobordism, Springer Monographs in Mathematic

(1998).

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12 STEFAN FRIEDL

[Sa] H. Sagan. Space-Filling Curves, Universitext, Springer-Verlag (1994)[Sv] N. Saveliev. Lectures on the topology of 3-manifolds. An introduction to the Casson invariant, Second

revised edition. de Gruyter Textbook (2012)[ST] H. Seifert and W. Threlfall. Lehrbuch der Topologie, Teubner Verlag (1934).[Se] J.-P. Serre. Trees, Springer-Verlag, Berlin-New York, 1980.[Se2] J.-P. Serre. A course in arithmetic, Graduate Texts in Mathematics 7. Springer-Verlag (1973).[Sm] S. Smale. Generalized Poincaré’s conjecture in dimensions greater than four, Annals of Mathematics.

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https://arxiv.org/abs/1610.00752

[Ta] C. Taubes. Gauge theory on asymptotically periodic 4-manifolds, J. Diff. Geom. 25 (1987), 363–430.[Tm] R. Thom. Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28:1 (1954),

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https://arxiv.org/abs/1610.00752http://canyon23.net/math/1985thesis.pdf


1. Topological spaces

1.1. The definition of a topological space. We recall the definition of a topologicalspace from Analysis IV.

Definition. A topological space is a pair (X, T ), where X is a set and T is a topology on X,i.e. T is a set of subsets of X with the following properties:

(1) the empty set and the entire set X are contained in T ,(2) the intersection of finitely many sets in T is again a set in T ,(3) the union of arbitrarily many sets in T is again a set in T .

The sets in T are called open.Example.

(1) Let (X, d) be a metric space. A subset U of X is called open if for every x ∈ Uthere exists an ϵ > 0 such that Bϵ(x) := {y ∈ X | d(x, y) < ϵ} is contained in U .We had already seen in Analysis II that

T := all open subsets of (X, d)is a topology on X. In the following we consider Rn as a metric space with theeuclidean metric and we always view Rn with the resulting topology, unless we sayexplicitly otherwise.

(2) Let X be a set, then T = {∅, X} is a topology on X. This topology is sometimescalled the trivial topology on X.

(3) Let X be a set and let T be the power set of X, i.e. the set of all subsets of X.Then T is also a topology on X. Put differently, T is the topology such that allsubsets are open. This topology is usually referred to as the discrete topology on X.

(4) Let X = R and let T be defined as follows:U ∈ T :⇐⇒ either U = ∅ or U is the complement of finitely many points in R.

For example the sets ∅,R\{π},R\{−1,√2} and also R (since it is the complement

of zero points) lie in T . It follows easily from elementary set theory that T is atopology on X = R.

(5) We consider the setX := Rn ∪ {∞},

i.e. X consists of Rn and an extra point ∞. We say U ⊂ X is open1, if both of thefollowing two conditions are satisfied:(a) for each point x ∈ U ∩ Rn there exists an ϵ > 0 such that Bϵ(x) ⊂ U ,(b) if ∞ ∈ U , then there exists a C > 0 such that {x ∈ Rn | ∥x∥ > C} ⊂ U .It is straightforward to see that this defines indeed a topology on X. For n = 1 wehad introduced this topological space in Analysis IV and we had referred to it asthe “line with a point at infinity”. We now refer to Rn ∪ {∞} as “Rn with a pointat infinity”.

1If we want to specify a topology, it suffices to specify which subsets are called “open”.

14 STEFAN FRIEDL

(6) We consider the set

X := R ∪ {∗},i.e. X consists of R and an extra point ∗. We say U ⊂ X is open, if the followingtwo conditions are satisfied:(a) for each point x ∈ U ∩ R there exists an ϵ > 0 such that (x− ϵ, x+ ϵ) ⊂ U ,(b) if ∗ ∈ U , then there exists an ϵ > 0 such that (−ϵ, 0) ∪ (0, ϵ) ⊂ U .We had seen in Analysis IV that this is indeed a topology on X. We refer to thistopological space as the “line with two zeros”.

(7) If (X, T ) is a topological space and if Y ⊂ X is a subset, then

S := {Y ∩ U |U ∈ T }

is a topology on Y . We refer to S as the subspace topology on Y . Unless we saysomething else we consider each subset Y of Rn always as a topological space withrespect to the subspace topology.

Now we recall several definitions from Analysis IV.

Definition. Let X be a topological space.2

(1) Let A ⊂ X be a subset. We say U ⊂ X is a neighborhood of A if there exists anopen set V such that A ⊂ V ⊂ U . We say U is an open neighborhood of A, if U isfurthermore open.

(2) We say X is Hausdorff, if given any two points x ̸= y there exist disjoint openneighborhoods U of x and V of y.

Example.(1) If X = R and A = [0, 2), then U = (−1, 3] and V = (−2,∞) are neighborhoods of

A in X.(2) We had already seen in Analysis II Proposition 1.8 that metric spaces are Hausdorff.

Furthermore we had seen in Analysis IV that the line with a point at infinity isalso Hausdorff and the same argument shows that Rn with a point at infinity isHausdorff. On the other hand we had seen in Analysis IV that the line with twozeros is not Hausdorff.

(3) A straightforward exercise shows that a topological space X is Hausdorff if and onlyif the diagonal D = {(x, x) |x ∈ X} is a closed subset of X ×X.

Definition. Let X be a topological space and let A be a subset of X.

(1) The interior◦A is defined as the union of all open sets of X that are contained in A.

(2) We say A is closed, if X \ A is open.(3) The closure A of A is defined as the intersection of all closed sets in X that contain A.

(4) The boundary of A in X is defined as ∂A := A \◦A.

2As usual we suppress the topology from the notation, i.e. we write “let X be a topological space”instead of the more precise “let (X, T ) be a topological space”.


Example. We consider X = R and A is the half-open interval [−1, 2). Then the interior of Ais the open interval (−1, 2) and the closure of A is the closed interval [−1, 2]. Furthermore∂A = {−1, 2}.

It follows immediately from the axioms of a topology that the interior of a set is an openset. Furthermore it is straightforward to see that the union of finitely many closed sets isagain closed and that the intersection of arbitrarily many closed sets is again closed. Itfollows easily that the closure of a subset is closed.

Definition. Let X be a topological space. An open covering of X is a family {Ui}i∈I ofopen subsets of X with

X =∪i∈I

Ui.

We say a topological space X is compact if for each open covering {Ui}i∈I of X there existfinitely many indices i1, . . . , ik ∈ I such that

X = Ui1 ∪ · · · ∪ Uik.Example. The Heine–Borel Theorem says that a subset A of Rn is compact if and only ifit is bounded and closed.

We recall the following well-known lemma.

Lemma 1.1. Let X be a topological space and let A ⊂ X be a compact subset. If X isHausdorff, then A is a closed subset of X.

For completeness’ sake we provide the proof.

Proof. Let X be a Hausdorff space and let A ⊂ X be a compact subset. We want to showthat X \ A is open. It suffices to prove the following claim.

Claim. Let x ∈ X \ A. Then there exists an open neighborhood V of x that is containedin X \ A.

We apply the Hausdorff-property to x and every y ∈ A. For every y ∈ A we obtaindisjoint open neighborhoods e Uy of y and Vy of x. Evidently we have

A =∪y∈A{y} ⊂

∪y∈A

(Uy ∩ A) ⊂ A.

Thus we see that {Uy ∩ A}y∈A is an open covering of A. Since A is compact there existy1, . . . , yk such that

A =k∪i=1

(Uyi ∩ A).

Now we consider

V :=k∩i=1

Vyi .

16 STEFAN FRIEDL

Since V is the intersection of finitely many open sets, it is open itself. Furthermore V doesnot intersect any of the Uyi , i = 1, . . . , k. Hence it V is disjoint from von A ⊂ Uy1∪· · ·∪Uyk .This concludes the proof of the claim. �Definition. We say a map f : X → Y between two topological spacesX and Y is continuous,if for each open set U in Y the preimage f−1(U) is open in X.

Example. We consider the set X = {A,B,C,D} where the topology is given by the setT := {∅, {A}, {C}, {D,A,B}, {B,C,D}, X}. Then it follows easily from the definitionsthat the map

S1 → X

eit 7→

A, if t ∈ (−π

4, π4),

B, if t = π4,

C if t ∈ (π4, 3π

4),

D if t = 3π4.

is continuous. This map is illustrated in Figure 1.

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fthe topology on X

A

B

C

D

Figure 1.

It is straightforward to see that the composition of two continuous maps is again con-tinuous. For maps between metric spaces we obtain the same notion of continuity as inAnalysis II.The following lemma states perhaps the most important feature of compact sets.

Lemma 1.2.

(1) Let f : X → Y be a continuous map. If X is compact, then f(X) is also compact.(2) Let f : X → R be a continuous map. If X is compact, then f assumes its maximum

and its minimum.(3) Let f : X → Y be a continuous map. If X is compact and if Y is Hausdorff, then

f(X) is a closed subset of Y .

Proof. In Analysis II we had proved the first two statements for metric spaces, the proof fortopological spaces is verbatim the same. The third statement is an immediate consequenceof Lemma 1.1 and the first statement. �Definition. We say a map f : X → Y between two topological spaces X and Y is a homeo-morphism if the following three properties are satisfied:


(1) f is continuous,(2) f is bijective,(3) f−1 : Y → X is also continuous.

If there exists a homeomorphism between X and Y we say that X and Y are homeomorphicand sometimes we write X ∼= Y .

The following proposition, that we had proved in Analysis IV, gives an often usefulcriterion for showing that a map is a homeomorphism.

Proposition 1.3. Let f : X → Y be a bijective continuous map between topological spaces.If X is compact and if Y is Hausdorff, then f is a homeomorphism.

Example. We consider the map

Φ: Sn → Rn ∪ {∞}

(x1, . . . , xn+1) 7→

{ (x1

1− xn+1, . . . ,

xn1− xn+1

), if xn+1 < 1,

∞, if xn+1 = 1.where we equip Rn∪{∞} with the topology that we had introduced on page 13. Outside ofthe “North pole” (0, . . . , 0, 1) this map is just the stereographic projection that is illustratedin Figure 2. This map is easily seen to be continuous3 and a bijection. Furthermore Sn

is compact by Heine-Borel and Rn ∪ {∞} is Hausdorff, as we had just pointed out above.Hence it follows from Proposition 1.3 that Φ is a homeomorphism.

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stereographic projection of P

North pole N = (0, 0, 1)

Pray emanating from N through P

Figure 2. Stereographic projection from S2 \ {(0, 0, 1)} onto R2.

Remark. If two topological spaces are homeomorphic, then they have the same topologicalproperties, i.e. they share all properties that are defined purely in terms of the topology.For example, if X and Y are homeomorphic, then X is Hausdorff if and only if Y isHausdorff, X is compact if and only if Y is compact and so on.

Convention. Henceforth any map between two topological spaces is assumed to be contin-uous, unless we say explicitly otherwise.

Definition.

3Is that really so easy?

18 STEFAN FRIEDL

(1) We say that a subset A ⊂ Rn is convex, if for any two distinct points P and Q in Athe segment PQ := {tP + (1− t)Q | t ∈ [0, 1]} also lies in A.

(2) Given a subset S of Rk the convex hull of S is defined as the intersection of allconvex subsets of Rk that contain S. Since the intersection of convex sets is againconvex we see that the convex hull of S is a convex subset of Rk.

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convex hull of Snot convex

P

Q

subset S of R2convex subset of R2

Figure 3.

Example. The convex hull of points P1, . . . , Pn ∈ Rk is easily seen to be given by the set{ n∑i=1

tiPi∣∣ t1, . . . , ti ∈ R≥0 and n∑

i=1

tn = 1}.

The following lemma gives a useful criterion for showing that subsets of Rn are homeo-morphic to an open or to a closed ball.

Lemma 1.4.

(1) Let A be a bounded open convex subset of Rn, then A is homeomorphic to the openn-dimensional ball Bn := {x ∈ Rn | ∥x∥ < 1}.

(2) Let A be a bounded closed convex subset of Rn such that the interior of A is non-empty. Then it follows that A is homeomorphic to the closed n-dimensional ballBn= {x ∈ Rn | ∥x∥ ≤ 1}. More precisely there exists a homeomorphism f : A→ Bn

with Φ(∂A) = Sn−1.

Examples.

(1) It follows from Lemma 1.4 that the open cube (0, 1)n is homeomorphic to Bn. Moregenerally, it follows from Lemma 1.4 that for any r, s ∈ N0 the product of ballsBr ×Bs ⊂ Rr × Rs = Rr+s is homeomorphic to Br+s.

(2) It follows from Lemma 1.4 that any triangle, i.e. any subset of R2 of the formA = {P + sv + tw | s, t ∈ [0, 1] and s + t ≤ 1} where P ∈ R2 and v, w are twolinearly independent vectors, is homeomorphic to B

2.

Proof.

(2) Let A be a bounded closed convex subset of Rn such that the interior of A is non-empty. After a translation we can assume that 0 lies in the interior of A. Given


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(0, 1)2 B2 B2

is homeomorphic to is homeomorphic to

Figure 4.

x ∈ A \ {0} we define

ρ(x) := sup{∥rx∥

∣∣ r ∈ R>0 and rx ∈ A}and

f(x) := x · ρ(x)∥x∥ .

Since A is closed we have f(x) ∈ A.

Claim. The map ρ : A→ R is positive, bounded and continuous.

Since A is bounded it follows that ρ is bounded. It follows from the definition ofρ that ρ(x) is always positive. Therefore it remains to show that ρ is continuous.Now let x ∈ A and let ϵ > 0. It suffices to show the following two statements:(a) there exists an open neighborhood U of x such that ρ(y) > ρ(x) − ϵ for all

y ∈ U ,(b) there exists an open neighborhood V of x such that ρ(y) < ρ(x) + ϵ for all

y ∈ V .We first show the existence of U . After possibly replacing ϵ by min{1

2ρ(x), ϵ} we

can suppose that ϵ ∈ (0, ρ(x)).Since 0 lies in the interior of A there exists an η > 0 such that Bη(0) ⊂ A. Since A

is convex, the convex hull C of Bη(0)∪{f(x)} is still contained4 in A. Furthermore,since Bη(0) is open it is straightforward to see that C

′ := C \ {f(x)} is an opensubset of Rn. We denote by Sn−1ρ(x)−ϵ the sphere of radius ρ(x) − ϵ around 0. Thepoint x · ρ(x)−ϵ∥x∥ lies on S

n−1ρ(x)−ϵ and it lies in C

′. Since C ′ is open there exists an open

neighborhood U ′ on Sn−1ρ(x)−ϵ that is contained in C′. We set

U := {rz | z ∈ U ′ and r ∈ (0, 1)}.

This is an open neighborhood of x and for any y ∈ U we have (ρ(x) − ϵ) · y ∈ A,i.e. for any y ∈ A we have ρ(y) ≥ ρ(x) − ϵ. Thus we have shown that the desiredneighborhood U exists. The existence of V is proved in a very similar way. We referto [Be, Chapter 11.3] for full details. This concludes the proof of the claim.

4Why is that the case?

20 STEFAN FRIEDL

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