Homology theory : an introduction to algebraic topology / by P.J. Hilton and S. Wylie.
Editor: Cambridge : University Press, 1960 (repr. 1965)Descripción: xv, 484 p. : il. ; 24 cmOtra clasificación: 55-01PART I. HOMOLOGY THEORY OF POLYHEDRA Background to Part I I.1 Analytic topology [3] I.2 Algebra [8] I.3 Zorn’s lemma [12] 1 The Topology of Polyhedra 1.1 Rectilinear simplexes [14] L2 Geometric simplicial complexes [17] 1.3 Polyhedra [20] 1.4 Regular subdivision [22] 1.5 The cone construction [25] 1.6 Homotopy [28] 1.7 Simplicial maps [34] 1.8 The simplicial approximation theorem [37] 1.9 Abstract simplicial complexes [41] 1.10* Infinite complexes [45] 1.11 Pseudodissections [49] 2 Homology Theory of a Simplicial Complex 2.1 Orientation of a simplex [53] 2.2 Chains, cycles and boundaries [56] 2.3 Homology groups [59] 2.4 H0(K) and connectedness [62] 2.5 Some examples and torsion [64] 2.6 Contrahomology and the Kronecker product [66] 2.7 Contrahomology examples [70] 2.8 Relative homology and contrahomology [73] 2.9 The exact sequences [80] 2.10 Homology groups of certain complexes [83] 2.11 Homology and contrahomology in infinite complexes [86] 2.12* Abstract cell complexes [87] 3 Chain Complexes 3.1 Chain and contrachain complexes [95] 3.2 Examples of chain complexes and chain maps [100] 3.3 Chain and contrachain homotopy [105] 3.4 Acyclic carriers [108] 3.5 Chain equivalences in simplicial complexes [113] 3.6 Continuous maps of polyhedra and the main theorems [116] 3.7 Local homology groups at a point of a polyhedron [124] 3.8 Simplex blocks [127] 3.9 Homology of real projective spaces [133] 3.10* Appendix on chain equivalence [136] 4 The Contrahomology Ring for Polyhedra 4.1 Definition of the ring for a complex [140] 4.2 Relativization, induced homomorphisms and topological invariance [145] 4.3 Calculations, examples and applications [149] 4.4* The cap product [153] 5 Abelian Groups and Homological Algebra 5.1 Standard bases for chain complexes [158] 5.2 Homology with general coefficients and contrahomology [167] 5.3 Free and divisible groups [178] 5.4 Homology and contrahomology in infinite complexes [183] 5.5 The products x, *, y, t [126] 5.6 Exact sequences [203] 5.7 Tensor products of chain complexes [209] 5.8 Appendix 1: Applications of the Hopf Trace Theorem [218] 5.9 Appendix 2: The group Ext (A, B) [220] 5.10 Appendix 3: Lens spaces [223] 6 The Fundamental Group and Covering Spaces 6.1 Definitions of the fundamental group [228] 6.2 Role of the base-point [232] 6.3 Calculation of the fundamental group of a polyhedron [235] 6.4 Further theorems and calculations [242] 6.5 Covering spaces [247] 6.6 Existence and uniqueness theorems for coyering spaces [253] 6.7 The universal covering space [261] 6.8 The covering space of a polyhedron [262] 6.9* Appendix: Fundamental group and covering groups of topological groups [265] PART H. GENERAL HOMOLOGY THEORY Background to Part II II. 1 Homotopy groups [273] II. 2 Function spaces and loop spaces [285] II. 3 Fibre spaces, relative homotopy groups and exact homotopy sequences [288] 7 Contrahomology and Maps 7.1 Introduction [290] 7.2 The obstruction contracycle [291] 7.3 The homotopy extension problem [294] 7.4 Applications [298] 7.5* Maps of polyhedra into Sm [303] 7.6 Local systems of groups and obstruction theory in non-simple spaces [307] 7.7* Contrahomology and compression [309] 8 Singular Homology Theory 8.1 Description and scope of the theory [313] 8.2 The normalized singular chain complex [318] 8.3 Cubical homology theory [321] 8.4 Equivalence theorems [324] 8.5 The properties of singular homology [329] 8.6 The singular homology theory of a polyhedron [336] 8.7 Homology groups of topological products [341] 8.8 The singular theory of n-connected spaces [344] 8.9* Singular homology with local coefficients [349] 8.10 Appendix: Cech contrahomology theory [353] 9 The Singular Contrahomology Ring 9.1 Definitions and properties [361] 9.2 Skew-commutativity of R*(X) [364] 9.3* Cup products in cubical contrahomology [367] 9.4 The contrahomology ring of a topological product [372] 9.5 The Hopf invariant [379] 9.6* Appendix: Naturality [387] 10* Spectral Homology Theory and Homology Theory of Groups 10.1 Filtration [394] 10.2 The spectral sequence of a differential filtered group [397] 10.3 Spectral theory for a differential filtered graded group [406] 10.4 Spectral theory of a map; fibre spaces [413] 10.5 Spectral contrahomology theory [422] 10.6 Spectral sequence of a fibre map: applications [428] 10.7 Homology and contrahomology of modules and groups [444] 10.8 The spectral sequence associated with a covering [464] 10.9 Appendix: Application to simplex blocks [469] 10.10 Appendix: The spectral sequence associated with a group, normal subgroup and quotient group [470]
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| 55 H656 An introduction to homotopy theory / | 55 H656a Algebraic topology : | 55 H656g General cohomology theory and K-theory / | 55 H656h Homology theory : | 55 H874 Homotopy theory / | 55 H874c Cohomology theory / | 55 H969-2 Fibre bundles / |
Bibliografía: p. 477.
PART I. HOMOLOGY THEORY OF POLYHEDRA --
Background to Part I --
I.1 Analytic topology [3] --
I.2 Algebra [8] --
I.3 Zorn’s lemma [12] --
1 The Topology of Polyhedra --
1.1 Rectilinear simplexes [14] --
L2 Geometric simplicial complexes [17] --
1.3 Polyhedra [20] --
1.4 Regular subdivision [22] --
1.5 The cone construction [25] --
1.6 Homotopy [28] --
1.7 Simplicial maps [34] --
1.8 The simplicial approximation theorem [37] --
1.9 Abstract simplicial complexes [41] --
1.10* Infinite complexes [45] --
1.11 Pseudodissections [49] --
2 Homology Theory of a Simplicial Complex --
2.1 Orientation of a simplex [53] --
2.2 Chains, cycles and boundaries [56] --
2.3 Homology groups [59] --
2.4 H0(K) and connectedness [62] --
2.5 Some examples and torsion [64] --
2.6 Contrahomology and the Kronecker product [66] --
2.7 Contrahomology examples [70] --
2.8 Relative homology and contrahomology [73] --
2.9 The exact sequences [80] --
2.10 Homology groups of certain complexes [83] --
2.11 Homology and contrahomology in infinite complexes [86] --
2.12* Abstract cell complexes [87] --
3 Chain Complexes --
3.1 Chain and contrachain complexes [95] --
3.2 Examples of chain complexes and chain maps [100] --
3.3 Chain and contrachain homotopy [105] --
3.4 Acyclic carriers [108] --
3.5 Chain equivalences in simplicial complexes [113] --
3.6 Continuous maps of polyhedra and the main theorems [116] --
3.7 Local homology groups at a point of a polyhedron [124] --
3.8 Simplex blocks [127] --
3.9 Homology of real projective spaces [133] --
3.10* Appendix on chain equivalence [136] --
4 The Contrahomology Ring for Polyhedra --
4.1 Definition of the ring for a complex [140] --
4.2 Relativization, induced homomorphisms and topological invariance [145] --
4.3 Calculations, examples and applications [149] --
4.4* The cap product [153] --
5 Abelian Groups and Homological Algebra --
5.1 Standard bases for chain complexes [158] --
5.2 Homology with general coefficients and contrahomology [167] --
5.3 Free and divisible groups [178] --
5.4 Homology and contrahomology in infinite complexes [183] --
5.5 The products x, *, y, t [126] --
5.6 Exact sequences [203] --
5.7 Tensor products of chain complexes [209] --
5.8 Appendix 1: Applications of the Hopf Trace Theorem [218] --
5.9 Appendix 2: The group Ext (A, B) [220] --
5.10 Appendix 3: Lens spaces [223] --
6 The Fundamental Group and Covering Spaces --
6.1 Definitions of the fundamental group [228] --
6.2 Role of the base-point [232] --
6.3 Calculation of the fundamental group of a polyhedron [235] --
6.4 Further theorems and calculations [242] --
6.5 Covering spaces [247] --
6.6 Existence and uniqueness theorems for coyering spaces [253] --
6.7 The universal covering space [261] --
6.8 The covering space of a polyhedron [262] --
6.9* Appendix: Fundamental group and covering groups of topological groups [265] --
PART H. GENERAL HOMOLOGY THEORY --
Background to Part II --
II. 1 Homotopy groups [273] --
II. 2 Function spaces and loop spaces [285] --
II. 3 Fibre spaces, relative homotopy groups and exact homotopy sequences [288] --
7 Contrahomology and Maps --
7.1 Introduction [290] --
7.2 The obstruction contracycle [291] --
7.3 The homotopy extension problem [294] --
7.4 Applications [298] --
7.5* Maps of polyhedra into Sm [303] --
7.6 Local systems of groups and obstruction theory in non-simple spaces [307] --
7.7* Contrahomology and compression [309] --
8 Singular Homology Theory --
8.1 Description and scope of the theory [313] --
8.2 The normalized singular chain complex [318] --
8.3 Cubical homology theory [321] --
8.4 Equivalence theorems [324] --
8.5 The properties of singular homology [329] --
8.6 The singular homology theory of a polyhedron [336] --
8.7 Homology groups of topological products [341] --
8.8 The singular theory of n-connected spaces [344] --
8.9* Singular homology with local coefficients [349] --
8.10 Appendix: Cech contrahomology theory [353] --
9 The Singular Contrahomology Ring --
9.1 Definitions and properties [361] --
9.2 Skew-commutativity of R*(X) [364] --
9.3* Cup products in cubical contrahomology [367] --
9.4 The contrahomology ring of a topological product [372] --
9.5 The Hopf invariant [379] --
9.6* Appendix: Naturality [387] --
10* Spectral Homology Theory and Homology Theory of Groups --
10.1 Filtration [394] --
10.2 The spectral sequence of a differential filtered group [397] --
10.3 Spectral theory for a differential filtered graded group [406] --
10.4 Spectral theory of a map; fibre spaces [413] --
10.5 Spectral contrahomology theory [422] --
10.6 Spectral sequence of a fibre map: applications [428] --
10.7 Homology and contrahomology of modules and groups [444] --
10.8 The spectral sequence associated with a covering [464] --
10.9 Appendix: Application to simplex blocks [469] --
10.10 Appendix: The spectral sequence associated with a group, normal subgroup and quotient group [470] --
MR, 22 #5963
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