Elements of Algebraic Topology

by Munkres,James R.

• ISBN13: 9780201627282
• ISBN10: 0201627280
• Format: Paperback
• Copyright: 1993-12-01
• Publisher: CRC Press

Table of Contents

Preface	vii
Homology Groups of a Simplicial Complex	1	(78)
Simplices	2	(5)
Simplicial Complexes and Simplicial Maps	7	(8)
Abstract Simplicial Complexes	15	(5)
Review of Abelian Groups	20	(6)
Homology Groups	26	(7)
Homology Groups of Surfaces	33	(8)
Zero-dimensional Homology	41	(2)
The Homology of a Cone	43	(4)
Relative Homology	47	(4)
Homology with Arbitrary Coefficients	51	(2)
The Computability of Homology Groups	53	(9)
Homomorphisms Induced by Simplicial Maps	62	(9)
Chain Complexes and Acyclic Carriers	71	(8)
Topological Invariance of the Homology Groups	79	(50)
Simplicial Approximations	80	(3)
Barycentric Subdivision	83	(6)
The Simplicial Approximation Theorem	89	(6)
The Algebra of Subdivision	95	(5)
Topological Invariance of the Homology Groups	100	(3)
Homomorphisms Induced by Homotopic Maps	103	(8)
Review of Quotient Spaces	111	(5)
Application: Maps of Spheres	116	(5)
Application: The Lefschetz Fixed-point Theorem	121	(8)
Relative Homology and the Eilenberg-Steenrod Axioms	129	(32)
The Exact Homology Sequence	129	(7)
The Zig-zag Lemma	136	(6)
Mayer-Vietoris Sequences	142	(3)
The Eilenberg-Steenrod Axioms	145	(4)
The Axioms for Simplicial Theory	149	(5)
Categories and Functors	154	(7)
Singular Homology Theory	161	(84)
The Singular Homology Groups	162	(6)
The Axioms for Singular Theory	168	(7)
Excision in Singular Homology	175	(8)
Acyclic Models	183	(3)
Mayer-Vietoris Sequences	186	(4)
The Isomorphism Between Simplicial and Singular Homology	190	(6)
Application: Local Homology Groups and Manifolds	196	(6)
Application: The Jordan Curve Theorem	202	(7)
More on Quotient Spaces	209	(5)
CW Complexes	214	(8)
The Homology of CW Complexes	222	(9)
Application: Projective Spaces and Lens Spaces	231	(14)
Cohomology	245	(54)
The Hom Functor	245	(6)
Simplicial Cohomology Groups	251	(6)
Relative Cohomology	257	(5)
Cohomology Theory	262	(8)
The Cohomology of Free Chain Complexes	270	(9)
Chain Equivalences in Free Chain Complexes	279	(2)
The Cohomology of CW Complexes	281	(4)
Cup Products	285	(7)
Cohomology Rings of Surfaces	292	(7)
Homology with Coefficients	299	(14)
Tensor Products	299	(8)
Homology with Arbitrary Coefficients	307	(6)
Homological Algebra	313	(54)
The Ext Functor	314	(6)
The Universal Coefficient Theorem for Cohomology	320	(7)
Torsion Products	327	(5)
The Universal Coefficient Theorem for Homology	332	(2)
Other Universal Coefficient Theorems	334	(4)
Tensor Products of Chain Complexes	338	(3)
The Kunneth Theorem	341	(9)
The Eilenberg-Zilber Theorem	350	(3)
The Kunneth Theorem for Cohomology	353	(7)
Application: The Cohomology Ring of a Product Space	360	(7)
Duality in Manifolds	367	(80)
The Join of Two Complexes	368	(6)
Homology Manifolds	374	(3)
The Dual Block Complex	377	(5)
Poincare Duality	382	(6)
Cap Products	388	(6)
A Second Proof of Poincare Duality	394	(5)
Application: Cohomology Rings of Manifolds	399	(9)
Application: Homotopy Classification of Lens Spaces	408	(6)
Lefschetz Duality	414	(10)
Alexander Duality	424	(2)
``Natural'' Versions of Lefschetz and Alexander Duality	426	(7)
Cech Cohomology	433	(12)
Alexander-Pontryagin Duality	445	(2)
Bibliography	447	(2)
Index	449