1 Introduction

What Are Manifolds？

Why Study Manifolds？

2 Topological Spaces

Topologies

Bases

Manifolds

Problems

3 New Spaces from 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

Homomorphisms Induced by Continuous Maps

Homotopy Equivalence

Higher Homotopy Groups

Categories and Functors

Problems

8 Circles and Spheres

The Fundamental Group of the Circle

Proofs of the Lifting Lemmas

Fundamental Groups of Spheres

Fundamental Groups of Product Spaces

Fundamental Groups of Manifolds

Problems

9 Some Group Theory

Free Products

Free Groups

Presentations of Groups

Free Abelian Groups

Problems

10 The Seifert-Van Kampen Theorem

Statement of the Theorem

Applications

Proof of the Theorem

Distinguishing Manifolds

Problems

11 Covering Spaces

Definitions and Basic Properties

Covering Maps and the Fundamental Group

The Covering Group

Problems

12 Classification of Coverings

Covering Homomorphisms

The Universal Covering Space

Proper Group Actions

The Classification Theorem

Problems

13 Homology

Singular Homology Groups

Homotopy Invariance

Homology and the Fundamental Group

The Mayer-Vietoris Theorem

Applications

The Homology of a Simplicial Complex

Cohomology

Problems

Appendix：Review of Prerequisites

Set Theory

Metric Spaces

Group Theory

References

Index