| Titre : | Topology |
| Auteurs : | Marco Manetti, Auteur |
| Type de document : | Monographie imprimée |
| Editeur : | Springer, 2023 |
| ISBN/ISSN/EAN : | 978-3-031-32141-2 |
| Format : | 1 vol. (391 p.) |
| Langues: | Anglais |
| Mots-clés: | metric and topological spaces ; compact and connected spaces ; sheaf cohomology |
| Résumé : |
This is an introductory textbook on general and algebraic topology, aimed at anyone with a basic knowledge of calculus and linear algebra. It provides full proofs and includes many examples and exercises. The covered topics include: set theory and cardinal arithmetic; axiom of choice and Zorn's lemma; topological spaces and continuous functions; con- nectedness and compactness; Alexandrov compactification; quotient topol- ogies; countability and separation axioms; prebasis and Alexander's theorem; the Tychonoff theorem and paracompactness; complete metric spaces and function spaces; Baire spaces; homotopy of maps; the fundamental group; the van Kampen theorem; covering spaces; Brouwer and Borsuk's theorems; free groups and free product of groups and basic category theory. While it is very concrete at the beginning, abstract concepts are gradually introduced. This second edition contains a new chapter with a topological introduction to sheaf cohomology and applications.It also corrects some inaccuracies and some additional exercises are proposed. The textbook is suitable for anyone needing a basic, comprehensive introduction to general and algebraic topology and its applications. |
| Sommaire : |
1 Geometrical Introduction to Topology ................................... 1 1.1 A Bicycle Ride Through the Streets of Rome ...................... 1 1.2 Topological Sewing .................................................. 4 1.3 The Notion of Continuity ............................................ 5 1.4 Homeomorphisms .................................................... 11 1.5 Facts Without Proof .................................................. 18 2 Sets ........................................................................... 21 2.1 Notations and Basic Concepts ....................................... 21 2.2 Induction and Completeness......................................... 24 2.3 Cardinality ............................................................ 26 2.4 The Axiom of Choice ................................................ 30 2.5 Zorn’s Lemma ........................................................ 33 2.6 The Cardinality of the Product....................................... 36 3 Topological Structures ..................................................... 41 3.1 Topological Spaces................................................... 42 3.2 Interior of a Set, Closure and Neighbourhoods ..................... 46 3.3 Continuous Maps..................................................... 49 3.4 Metric Spaces......................................................... 52 3.5 Subspaces and Immersions .......................................... 58 3.6 Topological Products................................................. 61 3.7 Hausdorff Spaces..................................................... 64 4 Connectedness and Compactness ......................................... 67 4.1 Connectedness........................................................ 68 4.2 Connected Components.............................................. 73 4.3 Covers................................................................. 76 4.4 Compact Spaces ...................................................... 77 4.5 Wallace’s Theorem ................................................... 81 4.6 Topological Groups .................................................. 84 4.7 Exhaustions by Compact Sets ....................................... 88 xi xii Contents 5 Topological Quotients ...................................................... 91 5.1 Identifications......................................................... 91 5.2 Quotient Topology ................................................... 94 5.3 Quotients by Groups of Homeomorphisms ......................... 96 5.4 Projective Spaces..................................................... 99 5.5 Locally Compact Spaces............................................. 103 5.6 The Fundamental Theorem of Algebra .......................... 105 6 Sequences.................................................................... 109 6.1 Countability Axioms ................................................. 109 6.2 Sequences............................................................. 113 6.3 Cauchy Sequences.................................................... 116 6.4 Compact Metric Spaces .............................................. 119 6.5 Baire’s Theorem ...................................................... 122 6.6 Completions ....................................................... 124 6.7 Function Spaces and Ascoli–Arzelà Theorem ................... 127 6.8 Directed Sets and Nets (Generalised Sequences) ............... 131 7 Manifolds, Infinite Products and Paracompactness .................... 135 7.1 Sub-bases and Alexander’s Theorem................................ 135 7.2 Infinite Products ...................................................... 137 7.3 Refinements and Paracompactness ............................... 139 7.4 Topological Manifolds ............................................... 142 7.5 Normal Spaces .................................................... 144 7.6 Separation Axioms ................................................ 146 8 More Topics in General Topology ..................................... 149 8.1 Russell’s Paradox..................................................... 149 8.2 The Axiom of Choice Implies Zorn’s Lemma ...................... 150 8.3 Zermelo’s Theorem .................................................. 153 8.4 Ultrafilters ............................................................ 156 8.5 The Compact-Open Topology ....................................... 158 8.6 Noetherian Spaces.................................................... 161 8.7 A Long Exercise: Tietze’s Extension Theorem ..................... 163 9 Intermezzo ............................................................... 167 9.1 Trees .................................................................. 167 9.2 Polybricks and Betti Numbers ....................................... 168 9.3 What Algebraic Topology Is......................................... 169 10 Homotopy ................................................................... 171 10.1 Locally Connected Spaces and the Functor π0 ..................... 171 10.2 Homotopy............................................................. 175 10.3 Retractions and Deformations ....................................... 179 10.4 Categories and Functors ............................................. 181 10.5 A Detour s ......................................................... 186 Contents xiii 11 The Fundamental Group .................................................. 187 11.1 Path Homotopy ....................................................... 187 11.2 The Fundamental Group ............................................. 193 11.3 The Functor π1 ....................................................... 195 11.4 The Sphere Sn Is Simply Connected (n ≥ 2) ....................... 198 11.5 Topological Monoids ............................................. 203 12 Covering Spaces ............................................................ 205 12.1 Local Homeomorphisms and Sections .............................. 205 12.2 Covering Spaces...................................................... 207 12.3 Quotients by Properly Discontinuous Actions ...................... 211 12.4 Lifting Homotopies .................................................. 214 12.5 Brouwer’s Theorem and Borsuk’s Theorem ........................ 219 12.6 A Non-Abelian Fundamental Group ................................ 223 13 Monodromy ................................................................. 225 13.1 Monodromy of Covering Spaces .................................... 225 13.2 Group Actions on Sets ............................................... 228 13.3 An Isomorphism Theorem ........................................... 230 13.4 Lifting Arbitrary Maps............................................... 233 13.5 Regular Coverings ................................................ 236 13.6 Universal Coverings .............................................. 239 13.7 Coverings with Given Monodromy .............................. 242 14 van Kampen’s Theorem.................................................... 245 14.1 van Kampen’s Theorem, Universal Version......................... 245 14.2 Free Groups........................................................... 250 14.3 Free Products of Groups ............................................. 254 14.4 Free Products and van Kampen’s Theorem ......................... 256 14.5 Attaching Spaces and Topological Graphs.......................... 260 14.6 Cell Complexes....................................................... 264 15 A Topological View of Sheaf Cohomology ............................... 267 15.1 Natural Transformations ............................................. 267 15.2 Sheaves ............................................................... 270 15.3 Exact Sequences...................................................... 279 15.4 Direct and Inverse Image Functors .................................. 286 15.5 Complexes of Abelian Groups....................................... 297 15.6 Cohomology of Sheaves ............................................. 303 15.7 Cohomology and Continuous Maps ................................. 310 15.8 Homotopy Invariance ................................................ 315 15.9 Cohomology of Spheres and Applications .......................... 319 16 Selected Topics in Algebraic Topology ................................ 325 16.1 Groupoids and Equivalence of Categories .......................... 325 16.2 Inner and Outer Automorphisms .................................... 328 16.3 The Cantor Set and Peano Curves ................................... 329 xiv Contents 16.4 The Topology of SO(3, R)........................................... 331 16.5 The Hairy Ball Theorem ............................................. 336 16.6 Complex Polynomial Functions ..................................... 337 16.7 Grothendieck’s Proof of van Kampen’s Theorem .................. 338 16.8 A Long Exercise: The Poincaré–Volterra Theorem................. 340 17 Hints and Solutions......................................................... 343 Reference .......................................................................... 371 Index ........................................................... |
Disponibilité (1)
| Cote | Support | Localisation | Statut |
|---|---|---|---|
| MAT/1085 | Livre | bibliothèque sciences exactes | Empruntable |
