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3 Burnside - Bldg. 2 Mathematics- Topology
3 Remote Warehouse Mathematics- Geometry and Trigonometry

A Combinatorial Introduction to Topology

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A Combinatorial Introduction to Topology Cover

 

Synopses & Reviews

Publisher Comments:

The creation of algebraic topology is a major accomplishment of 20th-century mathematics. The goal of this book is to show how geometric and algebraic ideas met and grew together into an important branch of mathematics in the recent past. The book also conveys the fun and adventure that can be part of a mathematical investigation.

Combinatorial topology has a wealth of applications, many of which result from connections with the theory of differential equations. As the author points out, "Combinatorial topology is uniquely the subject where students of mathematics below graduate level can see the three major divisions of mathematics — analysis, geometry, and algebra — working together amicably on important problems."

To facilitate understanding, Professor Henle has deliberately restricted the subject matter of this volume, focusing especially on surfaces because the theorems can be easily visualized there, encouraging geometric intuition. In addition, this area presents many interesting applications arising from systems of differential equations. To illuminate the interaction of geometry and algebra, a single important algebraic tool — homology — is developed in detail.

Written for upper-level undergraduate and graduate students, this book requires no previous acquaintance with topology or algebra. Point set topology and group theory are developed as they are needed. In addition, a supplement surveying point set topology is included for the interested student and for the instructor who wishes to teach a mixture of point set and algebraic topology. A rich selection of problems, some with solutions, are integrated into the text.

Book News Annotation:

Reprint, with corrections, of the work originally published by W.H. Freeman in 1979.
Annotation c. Book News, Inc., Portland, OR (booknews.com)

Synopsis:

Excellent text covers vector fields, plane homology and the Jordan Curve Theorem, surfaces, homology of complexes, more. Problems and exercises. Some knowledge of differential equations and multivariate calculus required.Bibliography. 1979 edition.

Synopsis:

The creation of algebraic topology is a major accomplishment of the 20th century. The goal of this book is to show how geometric and algebraic ideas met and grew together into an important branch of mathematics in the recent past. The book also conveys the fun and adventure that can be part of a mathematical investigation.

Synopsis:

Excellent text for upper-level undergraduate and graduate students shows how geometric and algebraic ideas met and grew together into an important branch of mathematics. Lucid coverage of vector fields, surfaces, homology of complexes, much more. Some knowledge of differential equations and multivariate calculus required. Many problems and exercises (some solutions) integrated into the text. 1979 edition. Bibliography.

Description:

Includes bibliographical references (p. [303]-304) and index.

Table of Contents

Chapter One Basic Concepts

1 The Combinatorial Method

2 Continuous Transformations in the Plane

3 Compactness and Connectedness

4 Abstract Point Set Topology

Chapter Two Vector Fields

5 A Link Between Analysis and Topology

6 Sperner's Lemma and the Brouwer Fixed Point Theorem

7 Phase Portraits and the Index Lemma

8 Winding Numbers

9 Isolated Critical Points

10 The Poincaré Index Theorem

11 Closed Integral Paths

12 Further Results and Applications

Chapter Three Plane Homology and Jordan Curve Theorem

13 Polygonal Chains

14 The Algebra of Chains on a Grating

15 The Boundary Operator

16 The Fundamental Lemma

17 Alexander's Lemma

18 Proof of the Jordan Curve Theorem

Chapter Four Surfaces

19 Examples of Surfaces

20 The Combinatorial Definition of a Surface

21 The Classification Theorem

22 Surfaces with Boundary

Chapter Five Homology of Complexes

23 Complexes

24 Homology Groups of a Complex

25 Invariance

26 Betti Numbers and the Euler Characteristic

27 Map Coloring and Regular Complexes

28 Gradient Vector Fields

29 Integral Homology

30 Torsion and Orientability

31 The Poincaré Index Theorem Again

Chapter Six Continuous Transformations

32 Covering Spaces

33 Simplicial Transformations

34 Invariance Again

35 Matrixes

36 The Lefschetz Fixed Point Theorem

37 Homotopy

38 Other Homologies

Supplement Topics in Point Set Topology

39 Cryptomorphic Versions of Topology

40 A Bouquet of Topological Properties

41 Compactness Again

42 Compact Metric Spaces

Hints and Answers for Selected Problems

Suggestions for Further Reading

Bibliography

Index

Product Details

ISBN:
9780486679662
Author:
Henle, Michael
Publisher:
Dover Publications
Author:
Henle
Author:
Mathematics
Location:
New York :
Subject:
Mathematics
Subject:
Algebra
Subject:
Geometry - General
Subject:
Algebra - General
Subject:
Topology
Subject:
Algebraic topology
Subject:
General Mathematics
Subject:
Mathematics-Geometry and Trigonometry
Copyright:
Edition Description:
Trade Paper
Series:
Dover Books on Mathematics
Series Volume:
bd. 100, nr. 2
Publication Date:
19940331
Binding:
TRADE PAPER
Language:
English
Illustrations:
Yes
Pages:
310
Dimensions:
8.5 x 5.38 in 0.77 pd

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Related Subjects


Science and Mathematics » Mathematics » Advanced
Science and Mathematics » Mathematics » Combinatorics
Science and Mathematics » Mathematics » Geometry » Geometry and Trigonometry
Science and Mathematics » Mathematics » Topology
Science and Mathematics » Physics » Crystallography

A Combinatorial Introduction to Topology New Trade Paper
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Product details 310 pages Dover Publications - English 9780486679662 Reviews:
"Synopsis" by ,
Excellent text covers vector fields, plane homology and the Jordan Curve Theorem, surfaces, homology of complexes, more. Problems and exercises. Some knowledge of differential equations and multivariate calculus required.Bibliography. 1979 edition.

"Synopsis" by , The creation of algebraic topology is a major accomplishment of the 20th century. The goal of this book is to show how geometric and algebraic ideas met and grew together into an important branch of mathematics in the recent past. The book also conveys the fun and adventure that can be part of a mathematical investigation.
"Synopsis" by ,
Excellent text for upper-level undergraduate and graduate students shows how geometric and algebraic ideas met and grew together into an important branch of mathematics. Lucid coverage of vector fields, surfaces, homology of complexes, much more. Some knowledge of differential equations and multivariate calculus required. Many problems and exercises (some solutions) integrated into the text. 1979 edition. Bibliography.

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