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

Por: Hilton, Peter JohnColaborador(es): Wylie, S. (Shaun), 1913-Editor: Cambridge : University Press, 1960 (repr. 1965)Descripción: xv, 484 p. : il. ; 24 cmOtra clasificación: 55-01
Contenidos:
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]
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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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