Algebraic topology recommendation

DOI link for Recommended Resources in Topology Allen Hatcher

Recommended Resources in Topology Allen Hatcher book

DOI link for Recommended Resources in Topology Allen Hatcher

Recommended Resources in Topology Allen Hatcher book

Edited ByKristine K. Fowler

As an undergrad, I loved abstract/modern algebra and graph theory. I find that I miss the experience of studying these areas of mathematics. It's a unique experience, as I'm sure many of you can relate. I'm looking for some books that can largely stand on their own and give a robust treatment of these fields. Thanks in advance.

It is somewhat jarring to hear of people who "know nothing about the homology theories of topological spaces and their applications" but are "familiar with homological algebra, category theory, spectral sequences (!!)" and the like. Certainly, this is a historically backwards position to be in, since a lot of these highly abstract theories with which you are familiar were almost entirely motivated by the concrete theory which you now wish to master.

Based on what you have said about your background, you will find Peter May's book "A Concise Course in Algebraic Topology" an appropriate read. Peter does not shy away from using categorical or homological machinery when dealing with this material, but also encourages his reader to become adept at the sort of calculations which yield insight into the nature of the subject. Amazingly, you can get the book freely off his website.

Also, to really hammer in the fact that you are engaging with a living, breathing, highly applicable subject, check out Kaczynski, Mischaikow and Mrozek's "Computational Homology" and Edelsbrunner and Harer's "Computational Topology" to find a breadth of applications of homology to physical and life sciences.

Update: The OP and others in a similar position may also be interested in my own upcoming book. You can find the cover here.

I don't see why I should not recommend my own book Topology and Groupoids (T&G) as a text on general topology from a geometric viewpoint and on 1-dimensional homotopy theory from the modern view of groupoids. This allows for a form of the van Kampen theorem with many base points, chosen according to the geometry of the situation, from which one can deduce the fundamental group of the circle, a gap in traditional accounts; also I feel it makes the theory of covering spaces easier to follow since a covering map of spaces is modelled by a covering morphism of groupoids. Also useful is the notion of fibration of groupoids. A further bonus is that there is a theorem on the fundamental groupoid of an orbit space by a discontinuous action of a group, not to be found in any other text, except a 2016 Bourbaki volume in French on "Topologie Algebrique": and that gives no example applications.

The book is available from amazon at $31.99 and a pdf version with hyperref and some colour is available from the web page for the book.

The book has no homology theory, so it contains only one initial part of algebraic topology.

BUT, another part of algebraic topology is in the new jointly authored book Nonabelian Algebraic Topology: filtered spaces, crossed complexes, cubical homotopy groupoids (NAT) published in 2011 by the European Mathematical Society. The print version is not cheap, but seems to me good value for 703 pages, and a pdf is available on my web page for the book. Motivation for the methods are given by a thorough presentation of the history and intuitions, and the book should be seen as a sequel to "Topology and Groupoids", to which it refers often.

The new book gives a quite different approach to the border between homotopy and homology, in which there is little singular homology, and no simplicial approximation. Instead, it gives a Higher Homotopy Seifert-van Kampen Theorem, which yields directly results on relative homotopy groups, including nonabelian ones in dimension 2 (!), and including generalisations of the Relative Hurewicz Theorem.

Part I, up to p. 204, is almost entirely on dimension 1 and 2, with lots of figures. You'll find little, if any, of the results on crossed modules in other algebraic topology texts. You will find relevant presentations on my preprint page.

Will this take on? The next 20 years may tell!

October 24, 2016 A new preprint Modelling and Computing Homotopy Types: I is available as an Introduction to the above NAT book. This expands on some material presented at CT2015, the Aveiro meeting on Category Theory.