Topology: A Geometric Approach
The book is a nice introduction to topology with an emphasis on its geometric aspects. It is written for two purposes. In the first part, consisting of three chapters, there is material suitable for a one semester basic course of topology. It starts with basic point set topology with special attention to topology of Rn, followed by a description of the classification of surfaces. The last topic introduced and studied in this part is the fundamental group of a space, including its application to surfaces and the vector field problem in the plane and on surfaces. To compute a fundamental group, the Seifert-van Kampen theorem is introduced and proved. The second part contains an extension of the material of the first part to the full-year course. It starts with the description of covering spaces, covering transformations and universal covering space, followed by a study of CW complexes and their properties from the homotopy point of view. Simplicial complexes and Δ-complexes for CW-complexes are described as well. The last chapter is devoted to the homology theory with special attention to its relations to homotopy. There are about 750 exercises of different levels, which can attract students to more active study of the subject. Solutions of selected exercises are included as an appendix (solutions to all exercises are available to the instructor in electronic form on application to Oxford University Press). The book is an excellent introduction to the subject and be recommend to anybody interested in geometry and topology as his/her first reading.