简介
this time of writing is the hundredth anniversary of the publication (1892) of poincare's first note on topology, which arguably marks the beginning of the subject of algebraic, or "combinatorial," topology. there was earlier scattered work by euler, listing (who coined the word "topology"), m/sbius and his band, riemann, klein, and betti. indeed, even as early as 1679, leibniz indicated the desirability of creating a geometry of the topological type. the establishment of topology (or "analysis situs" as it was often called at the time) as a coherent theory, however, belongs to poincar6.
curiously, the beginning of general topology, also called "point set
topology," dates fourteen years later when fr6chet published the first abstract treatment of the subject in 1906.
since the beginning of time, or at least the era of a'rchimedes, smoothmanifolds (curves, surfaces, mechanical configurations, the universe) havebeen a central focus in mathematics. they have always been at the core ofinterest in topology. after the seminal work of milnor, smale, and manyothers, in the last half of this century, the topological aspects of smoothmanifolds, as distinct from the differential geometric aspects, became a subject in its own right. while the major portion of this book is devoted to algebraic topology, i attempt to give the reader some glimpses into the beautiful and important realm of smooth manifolds along the way, and to instill the tenet that the algebraic tools are primarily intended for the understanding of the geometric world.
目录
preface
acknowledgments
chapter i general topology
1. metric spaces
2. topological spaces
3. subspaces
4. connectivity and components
5. separation axioms
6. nets (moore-smith convergence)
7. compactness
8. products
9. metric spaces again
10. existence of real valued functions
11. locally compact spaces
12. paracompact spaces
13. quotient spaces
14. homotopy
15. topological groups
16. convex bodies
17. the baire category theorem
.chapter ii differentiable manifolds
1. the implicit function theorem
2. differentiable manifolds
3. local coordinates
4. induced structures and examples
5. tangent vectors and differentials
6. sard's theorem and regular values
7. local properties of immersions and submersions
8. vector fields and flows
9. tangent bundles
10. embedding in euclidean space
11. tubular neighborhoods and approximations
12. classical lie groups
13. fiber bundles
14. induced bundles and whitney sums
15. transversality
16. thom-pontryagin theory
chapter iii fundamental group
1. homotopy groups
2. the fundamental group
3. covering spaces
4. the lifting theorem
5. the action of nl on the fiber
6. deck transformations
7. properly discontinuous actions
8. classification of covering spaces
9. the seifert-van kampen theorem
10. remarks on so(3)
chapter iv homology theory
1. homology groups
2. the zeroth homology group
3. the first homology group
4. functorial properties
5. homological algebra
6. axioms for homology
7. computation of degrees
8. cw-complexes
9. conventions for cw-complexes
10. cellular homology
11. cellular maps
12. products of cw-complexes
13. euler's formula
14. homology of real projective space
15. singular homology
16. the cross product
17. subdivision
18. the mayer-vietoris sequence
19. the generalized jordan curve theorem
20. the borsuk-ulam theorem
21. simplicial complexes
……
chapter v cohomology
chapter vi products and duality
chapter vii homotopy theory
appendices
bibliography
index of symbols
index
acknowledgments
chapter i general topology
1. metric spaces
2. topological spaces
3. subspaces
4. connectivity and components
5. separation axioms
6. nets (moore-smith convergence)
7. compactness
8. products
9. metric spaces again
10. existence of real valued functions
11. locally compact spaces
12. paracompact spaces
13. quotient spaces
14. homotopy
15. topological groups
16. convex bodies
17. the baire category theorem
.chapter ii differentiable manifolds
1. the implicit function theorem
2. differentiable manifolds
3. local coordinates
4. induced structures and examples
5. tangent vectors and differentials
6. sard's theorem and regular values
7. local properties of immersions and submersions
8. vector fields and flows
9. tangent bundles
10. embedding in euclidean space
11. tubular neighborhoods and approximations
12. classical lie groups
13. fiber bundles
14. induced bundles and whitney sums
15. transversality
16. thom-pontryagin theory
chapter iii fundamental group
1. homotopy groups
2. the fundamental group
3. covering spaces
4. the lifting theorem
5. the action of nl on the fiber
6. deck transformations
7. properly discontinuous actions
8. classification of covering spaces
9. the seifert-van kampen theorem
10. remarks on so(3)
chapter iv homology theory
1. homology groups
2. the zeroth homology group
3. the first homology group
4. functorial properties
5. homological algebra
6. axioms for homology
7. computation of degrees
8. cw-complexes
9. conventions for cw-complexes
10. cellular homology
11. cellular maps
12. products of cw-complexes
13. euler's formula
14. homology of real projective space
15. singular homology
16. the cross product
17. subdivision
18. the mayer-vietoris sequence
19. the generalized jordan curve theorem
20. the borsuk-ulam theorem
21. simplicial complexes
……
chapter v cohomology
chapter vi products and duality
chapter vii homotopy theory
appendices
bibliography
index of symbols
index
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