An Introduction to Algebraic Topology (Dover Books on Mathematics) Dover Ed Edition

This 1957 book by Andrew Hugh Wallace is an ideal easy first introduction to the most basic concepts of algebraic topology. Its concrete approach contrasts with the abstraction of some other books. In particular, there are no commutative diagrams, functors or category theory here.Anyone who has studied basic point-set topology will find the first three chapters yawn-worthy and archaic. So it's best to skip over them as quickly as possible. Chapter 4 presents the basic concepts of homotopy very concretely and simply. Just the concepts. No difficult theorems.Chapter 5 (or V) introduces simplexes, chains and homology groups. Here the first non-trivial observations are made in Section V.6 (pages 105-107), where the boundaries of simplexes and chains are defined, and it is shown that the boundary of the boundary of a simplex is empty. This is where the topology starts to become algebraic! In Section V.8, homologous cycles and homology groups are defined. (The homology relation is not clearly motivated at first, but it does soon become clear.)Chapter 7 ends with an excision theorem. Chapter 8 defines homology sequences (which are exact sequences) for pairs of topological manifolds. Chapter 9 introduces simplicial complexes. The difficulty of the algebra increases quite rapidly. So this is not a shallow book at all. But it always remains quite concrete so that the meanings of expressions can be determined without too much difficulty. Thus this books starts very easily indeed, but ends with some quite difficult concepts and theorems.

I bought this book because of its intuitive approach to Homology Theory.It is an excellent text, with many pictures. The author also is very clear on his statements.Excellent book, strongly recommended for those who, like me, want to learn the geometric ideas underlying to homology.

The text itself is fine, but the Kindle version is unusable. There are typos in formulas, things that simply make no sense grammatically or mathematically. This might be fine for reading a magazine article, but for a math text it make the material pretty much unusable: once you find a few typos, you don't know what to trust and what to try to reinterpret. Looks like it's too late to return it. I've ordered the hardcopy already, but I won't buy a Kindle version of a technical book again.

This book is written in a very old fashioned style --originally published in 1957.A little too wordy and not enough pictures to helpunderstand the material.There are many other books, such as Munkres,that I found a whole lot better.