Algebraic Topology
Hatcher, Allen
Algebraic Topology - New York: Cambridge University Press, 2001. - ix,544p.
Chapter 0. Some Underlying Geometric Notions 1 --
Homotopy and Homotopy Type 1 --
Cell Complexes 5 --
Operations on Spaces 8 --
Two Criteria for Homotopy Equivalence 10 --
Homotopy Extension Property 14 --
Chapter 1. Fundamental Group 21 --
1.1. Basic Constructions 25 --
Paths and Homotopy 25 --
Fundamental Group of the Circle 29 --
Induced Homomorphisms 34 --
1.2. Van Kampen's Theorem 40 --
Free Products of Groups 41 --
Van Kampen Theorem 43 --
Applications to Cell Complexes 50 --
1.3. Covering Spaces 56 --
Lifting Properties 60 --
Classification of Covering Spaces 63 --
Deck Transformations and Group Actions 70 --
1.A. Graphs and Free Groups 83 --
1.B. K(G,1) Spaces and Graphs of Groups 87 --
Chapter 2. Homology 97 --
2.1. Simplicial and Singular Homology 102 --
[Delta]-Complexes 102 --
Simplicial Homology 104 --
Singular Homology 108 --
Homotopy Invariance 110 --
Exact Sequences and Excision 113 --
Equivalence of Simplicial and Singular Homology 128 --
2.2. Computations and Applications 134 --
Degree 134 --
Cellular Homology 137 --
Mayer-Vietoris Sequences 149 --
Homology with Coefficients 153 --
2.3. Formal Viewpoint 160 --
Axioms for Homology 160 --
Categories and Functors 162 --
2.A. Homology and Fundamental Group 166 --
2.B. Classical Applications 169 --
2.C. Simplicial Approximation 177 --
Chapter 3. Cohomology 185 --
3.1. Cohomology Groups 190 --
Universal Coefficient Theorem 190 --
Cohomology of Spaces 197 --
3.2. Cup Product 206 --
Cohomology Ring 211 --
A Kunneth Formula 218 --
Spaces with Polynomial Cohomology 224 --
3.3. Poincare Duality 230 --
Orientations and Homology 233 --
Duality Theorem 239 --
Connection with Cup Product 249 --
Other Forms of Duality 252 --
3.A. Universal Coefficients for Homology 261 --
3.B. General Kunneth Formula 268 --
3.C. H-Spaces and Hopf Algebras 281 --
3.D. Cohomology of SO(n) 292 --
3.E. Bockstein Homomorphisms 303 --
3.F. Limits and Ext 311 --
3.G. Transfer Homomorphisms 321 --
3.H. Local Coefficients 327 --
Chapter 4. Homotopy Theory 337 --
4.1. Homotopy Groups 339 --
Definitions and Basic Constructions 340 --
Whitehead's Theorem 346 --
Cellular Approximation 348 --
CW Approximation 352 --
4.2. Elementary Methods of Calculation 360 --
Excision for Homotopy Groups 360 --
Hurewicz Theorem 366 --
Fiber Bundles 375 --
Stable Homotopy Groups 384 --
4.3. Connections with Cohomology 393 --
Homotopy Construction of Cohomology 393 --
Fibrations 405 --
Postnikov Towers 410 --
Obstruction Theory 415 --
4.A. Basepoints and Homotopy 421 --
4.B. Hopf Invariant 427 --
4.C. Minimal Cell Structures 429 --
4.D. Cohomology of Fiber Bundles 431 --
4.E. Brown Representability Theorem 448 --
4.F. Spectra and Homology Theories 452 --
4.G. Gluing Constructions 456 --
4.H. Eckmann-Hilton Duality 460 --
4.I. Stable Splittings of Spaces 466 --
4.J. Loopspace of a Suspension 470 --
4.K. Dold-Thom Theorem 475 --
4.L. Steenrod Squares and Powers 487 --
Topology of Cell Complexes 519 --
Compact-Open Topology 529.
9780521795401
Algebraic topology.
Topologia algebraiczna.
514.2 / HAT/A
Algebraic Topology - New York: Cambridge University Press, 2001. - ix,544p.
Chapter 0. Some Underlying Geometric Notions 1 --
Homotopy and Homotopy Type 1 --
Cell Complexes 5 --
Operations on Spaces 8 --
Two Criteria for Homotopy Equivalence 10 --
Homotopy Extension Property 14 --
Chapter 1. Fundamental Group 21 --
1.1. Basic Constructions 25 --
Paths and Homotopy 25 --
Fundamental Group of the Circle 29 --
Induced Homomorphisms 34 --
1.2. Van Kampen's Theorem 40 --
Free Products of Groups 41 --
Van Kampen Theorem 43 --
Applications to Cell Complexes 50 --
1.3. Covering Spaces 56 --
Lifting Properties 60 --
Classification of Covering Spaces 63 --
Deck Transformations and Group Actions 70 --
1.A. Graphs and Free Groups 83 --
1.B. K(G,1) Spaces and Graphs of Groups 87 --
Chapter 2. Homology 97 --
2.1. Simplicial and Singular Homology 102 --
[Delta]-Complexes 102 --
Simplicial Homology 104 --
Singular Homology 108 --
Homotopy Invariance 110 --
Exact Sequences and Excision 113 --
Equivalence of Simplicial and Singular Homology 128 --
2.2. Computations and Applications 134 --
Degree 134 --
Cellular Homology 137 --
Mayer-Vietoris Sequences 149 --
Homology with Coefficients 153 --
2.3. Formal Viewpoint 160 --
Axioms for Homology 160 --
Categories and Functors 162 --
2.A. Homology and Fundamental Group 166 --
2.B. Classical Applications 169 --
2.C. Simplicial Approximation 177 --
Chapter 3. Cohomology 185 --
3.1. Cohomology Groups 190 --
Universal Coefficient Theorem 190 --
Cohomology of Spaces 197 --
3.2. Cup Product 206 --
Cohomology Ring 211 --
A Kunneth Formula 218 --
Spaces with Polynomial Cohomology 224 --
3.3. Poincare Duality 230 --
Orientations and Homology 233 --
Duality Theorem 239 --
Connection with Cup Product 249 --
Other Forms of Duality 252 --
3.A. Universal Coefficients for Homology 261 --
3.B. General Kunneth Formula 268 --
3.C. H-Spaces and Hopf Algebras 281 --
3.D. Cohomology of SO(n) 292 --
3.E. Bockstein Homomorphisms 303 --
3.F. Limits and Ext 311 --
3.G. Transfer Homomorphisms 321 --
3.H. Local Coefficients 327 --
Chapter 4. Homotopy Theory 337 --
4.1. Homotopy Groups 339 --
Definitions and Basic Constructions 340 --
Whitehead's Theorem 346 --
Cellular Approximation 348 --
CW Approximation 352 --
4.2. Elementary Methods of Calculation 360 --
Excision for Homotopy Groups 360 --
Hurewicz Theorem 366 --
Fiber Bundles 375 --
Stable Homotopy Groups 384 --
4.3. Connections with Cohomology 393 --
Homotopy Construction of Cohomology 393 --
Fibrations 405 --
Postnikov Towers 410 --
Obstruction Theory 415 --
4.A. Basepoints and Homotopy 421 --
4.B. Hopf Invariant 427 --
4.C. Minimal Cell Structures 429 --
4.D. Cohomology of Fiber Bundles 431 --
4.E. Brown Representability Theorem 448 --
4.F. Spectra and Homology Theories 452 --
4.G. Gluing Constructions 456 --
4.H. Eckmann-Hilton Duality 460 --
4.I. Stable Splittings of Spaces 466 --
4.J. Loopspace of a Suspension 470 --
4.K. Dold-Thom Theorem 475 --
4.L. Steenrod Squares and Powers 487 --
Topology of Cell Complexes 519 --
Compact-Open Topology 529.
9780521795401
Algebraic topology.
Topologia algebraiczna.
514.2 / HAT/A