This book is an introduction to manifolds at the beginning graduate level:It contains the essential topological ideas that are needed for the furtherstudy of manifolds, particularly in the context of differential geometry,algebraic topology, and related fields￡?Its guiding philosophy is to develop these ideas rigorously but economically, with minimal prerequisites and plenty of geometric intuition. Here at the University of Washington, for example, this text is used for the first third of a year-long course on the geometry and topology of anifolds; the remaining two-thirds focuses on smooth manifolds.

Preface

1 Introduction

What Are Manifolds?

Why Study Manifolds?

2 Topologiacl Spaces

Topologies

Bases

Manifolds

Problems

3 New Spaces form Old

Subspaces

Product Spaces

Quotient Spaces

Group Actions

Problems

4 Connectedness and Compactness

Connectedness

Compactness

Locally Compact Hausdorff Spaces

Problems

5 Simplicial Complexes

Euclidean Simplicial Complexes

Abstract Simplicial Complexes

Triangulation Theorems

Orientations

Combinatorial Invariants

Problems

6 Curves and Surfaces

Classification of Curves

Surfaces

Connected Sums

Polygonal Presentations of Surfaces

Classification of Surface Presentations

Combinatorial Invariants

Problems

7 Homotopy and the Fundamental Group

Homotopy

The Fundamental Group

Homomorphisma Induced by Continuous Maps

……

8 Circles and Spheres

9 Some Group Theory

10 The Seifert-Van Kampen Theorem

11 Covering Spaces

12 Classification of Coverings

13 Homology

Appendix：Review of Prerequisites

References

Index