A Geometric Introduction to Topology

This book is a brief introduction to algebraic topology and is written by one of the major contributors to the subject. Written for undergraduates, it does not presuppose any background in topology, and the author concentrates strictly on subsets of Euclidean space. And, interestingly, the author does not introduce homology and cohomology using simplicial complexes, but instead uses the Cech theory and singular homology. Also, and somewhat disappointingly, the fundamental group is not discussed at all. The author is very concrete in his presentation, and he includes effective sets of exercises at the end of each chapter. He also introduces the necessary algebra at various places in the book. Some of the highlights in the book include: 1. The discussion of the zeroth cohomology group of a topological space, which is introduced as the collection of continuous maps from the space into the integers. This of course is what is called singular cohomology, and the author shows how it is related to the path components of the space, one consequence being that if there is only one path component, then the zeroth cohomology group has only constant maps. The singular homology group is then defined as the free group on the path component space. 2. The treatment of homotopy, and how it is related to the first singular cohomology group, the latter being the collection of maps from a space to the unit circle. The author also gives an interesting exercise dealing with quaternions. 3. The study of the algebraic topology of the unit circle. This discussion introduces the important concept of the degree of a map, and this is used to prove the fundamental theorem of algebra and the Brouwer fixed point theorem in the plane. 4. The treatment of the Mayer-Vietoris theorem. This is a fundamental result in algebraic topology, and the author computes the first singular cohomology group of a product as an example of this result. 5. The discussion on duality, which, at this level, is a very original presentation that essentially relies on extending Eilenberg's criterion on separation of points by compact plane sets.