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[null,"1.","Record Nr.",null,"UNISA996466400803316"]
[null,null,"Autore",null,"Bray Clark"]
[null,null,"Titolo",null,"Algebraic Topology"]
[null,null,"Pubbl/distr/stampa",null,"Cham : , : Springer International Publishing AG, , 2021"]
[{"$nulls":4},"©2021"]
[null,null,"ISBN",null,"3-030-70608-7"]
[null,null,"Descrizione fisica",null,"1 online resource (216 pages)"]
[null,null,"Soggetti",null,"Topologia algebraica"]
[{"$nulls":4},"Llibres electrònics"]
[null,null,"Lingua di pubblicazione",null,"Inglese"]
[null,null,"Formato",null,"Materiale a stampa"]
[null,null,null,"Livello bibliografico","Monografia"]
[null,null,"Nota di contenuto",null,"Intro -- Foreword -- Introduction -- Contents -- 1 Surface Preliminaries -- 1.1 Surfaces -- 1.2 Euclidean Space -- 1.3 Open Sets -- 1.4 Functions and Their Properties -- 1.5 Continuity -- 1.6 Problems -- 2 Surfaces -- 2.1 The Definition of a Surface -- 2.2 Examples of Surfaces -- 2.3 Spheres as Surfaces -- 2.4 Surfaces with Boundary -- 2.5 Closed, Bounded, and Compact Surfaces -- 2.6 Equivalence Relations and Topological Equivalence -- 2.7 Homeomorphic Spaces -- 2.8 Invariants -- 2.9 Problems -- 3 The Euler Characteristic and Identification Spaces -- 3.1 Triangulations and the Euler Characteristic -- 3.2 Invariance of the Euler Characteristic -- 3.3 Identification Spaces -- 3.4 ID Spaces as Surfaces -- 3.5 Abstract Topological Spaces -- 3.6 The Quotient Topology -- 3.7 Further Examples of ID Spaces -- 3.8 Triangulations of ID Spaces -- 3.9 The Connected Sum -- 3.10 The Euler Characteristic of a Compact Surface with Boundary -- 3.11 Problems -- 4 Classification Theorem of Compact Surfaces -- 4.1 The Geometry of the Projective Plane and the Klein Bottle -- 4.2 Orientable and Nonorientable Surfaces -- 4.3 The Classification Theorem for Compact Surfaces -- 4.4 Compact Surfaces Have Finite Triangulations -- 4.5 Proof of the Classification Theorem -- 4.6 Problems -- 5 Introduction to Group Theory -- 5.1 Why Use Groups? -- 5.2 A Motivating Example -- 5.3 Definition of a Group -- 5.4 Examples of Groups -- 5.5 Free Groups, Generators, and Relations -- 5.6 Free Products -- 5.7 Problems -- 6 Structure of Groups -- 6.1 Subgroups -- 6.2 Direct Products of Groups -- 6.3 Homomorphisms -- 6.4 Isomorphisms -- 6.5 Existence of Homomorphisms -- 6.6 Finitely "]
[{"$nulls":4},"Generated Abelian Groups -- 6.7 Problems -- 7 Cosets, Normal Subgroups, and Quotient Groups -- 7.1 Cosets -- 7.2 Lagrange's Theorem and Its Consequences -- 7.3 Coset Spaces and Quotient Groups."]
[{"$nulls":4},"7.4 Properties and Examples of Normal Subgroups -- 7.5 Coset Representatives -- 7.6 A Quotient of a Dihedral Group -- 7.7 Building up Finite Groups -- 7.8 An Isomorphism Theorem -- 7.9 Problems -- 8 The Fundamental Group -- 8.1 Paths and Loops on a Surface -- 8.2 Equivalence of Paths and Loops -- 8.3 Equivalence Classes of Paths and Loops -- 8.4 Multiplication of Path and Loop Classes -- 8.5 Definition of the Fundamental Group -- 8.6 Problems -- 9 Computing the Fundamental Group -- 9.1 Homotopies of Maps and Spaces -- 9.2 Computing the Fundamental Group of a Circle -- 9.3 Problems -- 10 Tools for Fundamental Groups -- 10.1 More Fundamental Groups -- 10.2 The Degree of a Loop -- 10.3 Fundamental Group of a Circle-Redux -- 10.4 The Induced Homomorphism on Fundamental Groups -- 10.5 Retracts -- 10.6 Problems -- 11 Applications of Fundamental Groups -- 11.1 The Fundamental Theorem of Algebra -- 11.2 Further Applications of the Fundamental Group -- 11.3 Problems -- 12 The Seifert-Van Kampen Theorem -- 12.1 Wedges of circles -- 12.2 The Seifert-Van Kampen Theorem: First Version -- 12.3 More Fundamental Groups -- 12.4 The Seifert-Van Kampen Theorem: Second Version -- 12.5 The Fundamental Group of a Compact Surface -- 12.6 Even More Fundamental Groups -- 12.7 Proof of the Second Version of the Seifert-Van Kampen Theorem -- 12.8 General Seifert-Van Kampen Theorem -- 12.9 Groups as Fundamental Groups -- 12.10 Problems -- 13 Introduction to Homology -- 13.1 The Idea of Homology -- 13.2 Chains -- 13.3 The Boundary Map -- 13.4 Homology -- 13.5 The Zeroth Homology Group -- 13.6 Homology of the Klein Bottle -- 13.7 Homology and Euler Characteristic -- 13.8 Homology and Orientability -- 13.9 Smith Normal Form -- 13.10 The Induced Map on Homology -- 13.11 Problems -- 14 The Mayer-Vietoris Sequence -- 14.1 Exact Sequences -- 14.2 The Mayer-Vietoris Sequence."]
[{"$nulls":4},"14.3 Homology of Orientable Surfaces -- 14.4 The Jordan Curve Theorem -- 14.5 The Hurewicz Map -- 14.6 Problems -- Correction to: The Seifert-Van Kampen Theorem -- Correction to:  Chapter 12 in: C. Bray et al., Algebraic Topology, https://doi.org/10.1007/978-3-030-70608-112 -- Appendix A Topological Notions -- A.1  Compactness Results -- A.2  Technical Conditions for Abstract Surfaces -- Appendix B A Brief Look at Singular Homology -- Appendix C Hints for Selected Problems -- Appendix  References --  -- Index."]