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Homology theory : an introduction to algebraic topology / by P.J. Hilton and S. Wylie.

By: Hilton, Peter John.
Contributor(s): Wylie, S. (Shaun), 1913-.
Material type: materialTypeLabelBookPublisher: Cambridge : University Press, 1960 (repr. 1965)Description: xv, 484 p. : il. ; 24 cm.Other classification: 55-01
Contents:
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] --
List(s) this item appears in: Textos de topología algebraica (MSC 55-01)
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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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