Elements of Algebraic Topology

Auteur

James Munkres served on the MIT Mathematics Faculty from 1960-2000, and continues as Senior Lecturer. He received the PhD in Mathematics from the University of Michigan under the supervision of Edwin Moise in 1956. Professor Munkres is a differential topologist and is also responsible for the Munkres assignment algorithm. He authored numerous texts. His distinctions include the MIT School of Science Teaching Prize for Undergraduate Education in 1984, and an Honorary Doctorate from Nebraska Wesleyan University in 1989.

Steven G. Krantz is a professor at Washington University in St. Louis where he teaches mathematics. He received his Ph.D. from Princeton University and since then has taught at UCLA, Princeton University, and Pennsylvania State University. Dr. Krantz has written over 175 scholarly papers and more than 65 books. He is the founding editor of the Journal of Geometric Analysis. He was named a fellow of the American Mathematical Society and has received the Chauvenet Prize, Beckenbach Book Award, and Kemper Prize.

Harold R. Parks received his Ph.D. in mathematics from Princeton University. He was a J. D. Tamarkin Instructor at Brown University. He then assumed a tenure-track position at Oregon State University. He was promoted to Professor in 1989. He spent the academic year 1982-83 at Indiana University as a Visiting Associate Professor. During his time in the Mathematics Department of Oregon State, he served at various times as Assistant Department Chair, Associate Department Chair, and Department Chair. He has written 8 books and 42 scholarly papers. He edits 2 journals. He is an AMS Fellow.

Texte du rabat

With coverage of homology and cohomology theory, universal coefficient theorems, Kunneth theorem, duality in manifolds, and applications to classical theorems of point-set topology, this book is perfect for communicating complex topics and the fun nature of algebraic topology for beginners.

Contenu

1 Homology Groups of a Simplicial Complex

1.1 Introduction

1.2 Simplices

1.3 Simplicial Complexes and Simplicial Maps

1.4 Abstract Simplicial Complexes

1.5 Review of Abelian Groups

1.6 Homology Groups

1.7 Homology Groups of Surfaces

1.8 Zero-Dimensional Homology

1.9 The Homology of a Cone

1.10 Relative Homology

1.11 Homology with Arbitrary Coefficients

1.12 The Computability of Homology Groups

1.13 Homomorphisms Induced by Simplicial Maps

1.14 Chain Complexes and Acyclic Carriers

2 Topological Invariance of the Homology Groups

2.1 Introduction

2.2 Simplicial Approximations

2.3 Barycentric Subdivision

2.4 The Simplicial Approximation Theorem

2.5 The Algebra of Subdivision

2.6 Topological Invariance of the Homology Groups

2.7 Homomorphisms Induced by Homotopic Maps

2.8 Review of Quotient Spaces

2.9 Application: Maps of Spheres

2.10 The Lefschetz Fixed Point Theorem

3 Relative Homology and the Eilenberg-Steenrod Axioms

3.1 Introduction

3.2 The Exact Homology Sequence

3.3 The Zig-Zag Lemma

3.4 The Mayer-Vietoris Sequence

3.5 The Eilenberg-Steenrod Axioms

3.6 The Axioms for Simplicial Theory

3.7 Categories and Functors

4 Singular Homology Theory

4.1 Introduction

4.2 The Singular Homology Groups

4.3 The Axioms for Singular Theory

4.4 Excisionin Singular Homology

4.5 Acyclic Models

4.6 Mayer-Vietoris Sequences

4.7 The Isomorphism Between Simplicial and Singular Homology

4.8 Application: Local Homology Groups and Manifolds

4.9 Application: The Jordan Curve Theorem

4.10 The Fundamental Group

4.11 More on Quotient Spaces

4.12 CW Complexes

4.13 The Homology of CW Complexes

4.14 Application: Projective Spaces and Lens Spaces

5 Cohomology

5.1 Introduction

5.2 The Hom Functor

5.3 Simplicial Cohomology Groups

5.4 Relative Cohomology

5.5 Cohomology Theory

5.6 The Cohomology of Free Chain Complexes

5.7 Chain Equivalences in Free Chain Complexes

5.8 The Cohomology of CW Complexes

5.9 Cup Products

5.10 Cohomology Rings of Surfaces

6 Homology with Coefficients

6.1 Introduction

6.2 Tensor Products

6.3 Homology with Arbitrary Coefficients

7 Homological Algebra

7.1 Introduction

7.2 The Ext Functor

7.3 The Universal Coefficient Theorem

7.4 Torsion Products

7.5 The Universal Coefficient Theorem for Homology

7.6 Other Universal Coefficient Theorems

7.7 Tensor Products of Chain Complexes

7.8 The Künneth Theorem

7.9 TheEilenberg-Zilber Theorem

7.10 The Künneth Theorem for Cohomolgy

7.11 Application: The Cohomology Ring of a Product Space

8 Duality in Manifolds

8.1 Introduction

8.2 The Join of Two Complexes

8.3 Homology Manifolds

8.4 The Dual Block Complex

8.5 Poincaré Duality

8.6 Cap Products

8.7 A Second Proof of Poincaré Duality

8.8 Application: Cohomology Rings of Manifolds

8.9 Application: Homotopy Classification of Lens Spaces

8.10 Lefschetz Duality

8.11 Alexander Duality

8.12 Natural Versions of Duality

8.13 ech Cohomology

8.14 Alexander-Pontryagin Duality