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Algebraic topologyShowing 8 featured editions. View all 8 editions?
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1
Algebraic Topology: An Introduction (Graduate Texts in Mathematics)
January 8, 1990, Springer
in English
0387902716 9780387902715
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2
Algebraic topology, an introduction
1977, Springer-Verlag, Springer
in English
- 4th corrected printing.
0387902716 9780387902715
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WorldCat
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Book Details
First Sentence
"This textbook is designed to introduce advanced undergraduate or beginning graduate students to algebraic topology as painlessly as possible."
Table of Contents
Preface
Note to the Student
Contents
One. Two-Dimensional Manifolds
1. Introduction
2. Definition and examples of n-manifolds
3. Orientable vs. nonorientable manifolds
4. Examples of compact, connected 2-manifolds
5. Statement of the classification theorem for compact surfaces
6. Triangulation of compact surfaces
7. Proof of Theorem 5.1
8. The Euler characteristic of a surface
9. Manifolds with Boundary
10. The classification of compact, connected 2-manifolds with boundary
11. The Euler characteristic of a bordered surface
12. Models of compact bordered surfaces in Euclidean 3-space
13. Remarks on noncompact surfaces
Two. The Fundamental Group
1. Introduction
2. Basic Notation and Terminology
3. Definition of the fundamental group of a space
4. The effect of a continuous mapping on the fundamental group
5. The fundamental group of a circle is infinite cyclic
6. Application: The Brouwer fixed-point theorem in dimension 2
7. The fundamental group of a product space
8. Homotopy type and homotopy equivalence of spaces
Three. Free Groups and Free Products of Groups
1. Introduction
2. The weak product of abelian groups
3. Free abelian groups
4. Free products of groups
5. Free groups
6. The presentation of groups be generators and relations
7. Universal mapping problems
Four. Seifert and Van Kampen Theorem on the Fundamental Group of the Union of Two Spaces. Applications
1. Introduction
2. Statement and proof of the theorem of Seifert and Van Kampen
3. First application of Theorem 2.1
4. Second application of Theorem 2.1
5. Structure of the fundamental group of a compact surface
6. Application to knot theory
Five. Covering Spaces
1. Introduction
2. Definition and some examples of covering spaces
3. Lifting of paths to a covering space
4. The fundamental group of a covering space
5. Lifting of arbitrary maps to a covering space
6. Homomorphisms and automorphisms of covering spaces
7. The action of the group $pi(X,x)$ on the set $p^{-1}(x)$
8. Regular covering spaces and quotient spaces
9. Application: The Borsuk-Ulam theorem for the 2-sphere
10. The existence theorem for covering spaces
11. The induced covering space over a subspace
12. Point set topology of covering spaces
Six. The Fundamental Group and Covering Spaces of a Graph. Applications to Group Theory
1. Introduction
2. Definition and Examples
3. Basic properties of graphs
4. Trees
5. The fundamental group of a graph
6. The Euler characteristic of a finite graph
7. Covering spaces of a graph
8. Generators for a subgroup of free group
Seven. The Fundamental Group of Higher Dimensional Spaces
1. Introduction
2. Adjunction of 2-cells to a space
3. Adjunction of higher dimensional cells to a space
4. CW-complexes
5. The Kurosh subgroup theorem
6. Grushko's theorem
Eight. Epilogue
Appendix A. The Quotient Space of Identification Space Topology
1. Definition and basic properties
2. A generalization of the quotient space topology
3. Quotient spaces and product spaces
4. Subspace of a quotient space vs. quotient space of a subspace
5. Conditions for a quotient space to be a Hausdorff space
Appendix B. Permutation Groups or Transformation Groups
1. Basic definitions
2. Homogeneous G-spaces
Index
Edition Notes
ID Numbers
First Sentence
"This textbook is designed to introduce advanced undergraduate or beginning graduate students to algebraic topology as painlessly as possible."
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