Elementary Concepts of Topology

This book introduces the algebraic machinery of homology theory and uses it to prove the invariance of dimension and the invariance of Betti numbers. With this point of view, manifolds should be taken to be complexes, and their properties should be studied in terms of linear combinations of its simplexes. So, for instance, in an appropriately oriented tetrahedron the "sum" of the faces is zero since each edge is counted twice with opposite orientation. But if we try to do the same thing in the projective plane or on a Möbius strip we will find that the summation of all faces leaves a boundary, revealing the difference between orientable and nonorientable surfaces. The natural algebraic equivalence of cycles of edges in a complex (homology) is close enough to topological equivalence (homotopy) to make the notion useful; in particular, the connectivity of a surface determines the number of generators of the free part of its homology group (the Betti number), so invariance of Betti numbers does give us useful topological information. Of course, if it was only for surfaces, homology and Betti numbers would be plainly inferior to the fundamental group, and indeed Alexandroff plays down surface topology while offering little in return except unsubstantiated reassurance that "anybody who wants to study topology for the sake of its applications must begin with the Betti groups".

A very slender book, but it sets out the basic ideas of algebraic topology and HOW THEY RELATE TO EACH OTHER. A roadmap to what all this simplex stuff is all about.For sophisticated mathematical readers only. Perfect adjunct to any first course.(And a "lemniscate" is a figure 8).

A great book, born in a great moment of mathematics. Alexandroff explains, and shows in pictures, what topology is basically about and why "homology groups" are the way to do it. Anyone can follow this who has had multivariable calculus, plus seen the definition of a group (as in, say, arithmetic modulo 2). In 55 profusely illustrated yet rigorous pages Alexandroff shows how to define topological manifolds, cut them into "simplices", and keep track of simplices algebraically. He proves the two founding theorems of topology: the dimension of manifolds, and their homology groups, are both preserved by topological isomorphisms.Alexandroff was a favorite student of Emmy Noether, and L.E.J. Brouwer, and followed Hilbert's lectures. The greatest algebraist, the greatest topologist, and the greatest mathematician of the early 20th century all had direct input into this book. All believed the most important, deepest mathematics can be made the clearest. They were right.

This book is perfect for the advanced undergraduate--if you've taken modern algebra (groups, rings, fields, etc.), real analysis, some point-set topology, and are curious about algebraic topology, then this little book is time and money well spent. In about 50 relatively easy-to-read pages, you'll be able to sit down with your favorite topological spaces and actually do homological calculations. One of the book's main appeals is that it was originally published in 1932--well before homological algebra (aka "diagram chasing") obscured the beauty of algebraic topology.