# A toolbox for algebraic topology

This question has a very general part and a rather concrete part.

General:

When one wants to prove something in algebraic topology (actually in all parts of mathematics) one obviously needs some good ideas, but first one has to have a good set of tools at hand. Introductory books in algebraic topology provide a number of such tools like long exact sequences to name just one. If one proceeds working in that field and reaches research level more and more tools are just treated as "common knowledge". They are used in papers according to the current situation and often left without quotation.

Over the time one gathers plenty of those tools, but I for my part still take many of them as black boxes. When I use them I always have the feeling of walking on very thin ice. Most advanced books have some of those tools scattered in their body and finding a particular one is often harder than it should be. There they are used to build up a certain theory and often don't reveal themselves as useful tools with applications beyond the topic of the respective book. Moreover it is one thing to find the reference for a statement one knows to be more or less true, but realising which tool one has to use when one isn't even aware of the precise statement is a different story.

So the first question:

Are there any good books which provide a box of tools used in modern algebraic topology? Maybe something like "AT for the working mathematician".

They should come with a proof but not necessarily with applications (for the above reason).

Special:

The above is incredibly imprecise and there are so many ways to interpret the question. Hence one example of a statement I actually want to know about, which might also give a hint at what I am looking for.

Second question: What is the precise statement/where can I find a proof

Given a commutative square of fibrations (cofibrations). Then the fibers (cofibers) in the horizontal direction are homotopy equivalent if and only if the fibers (cofibers) in the vertical direction are homotopy equivalent.

The square is then cartesian, cocartesian, bicartesian?

Edit 1: Now that I think about it it looks like the second question is just an application of the snake lemma. I have to work out the details. Still this statement may stand as an example of what I am looking for.

Edit 2: A book which seems to go in the direction of what I describe might be Goerss/Jardine: simplicial homotopy theory.

• If you take the phrase "AT for the working mathematician" and "applications" at face value, you should include computational topology, e.g. computational Morse theory and persistent homology. There are a couple of books on these by Edelsbrunner/Harer and Zomorodian that are pretty good. Commented Jul 13, 2012 at 20:34
• The second question is discussed a bit in the book-in-progress of Munson and Volic. I think it might be stated without a proof in the last version I saw... But it's not hard to prove if you use the right models. I needed a version in which the spaces might be disconnected (not a big deal, really). See p. 30 here: arxiv.org/abs/1206.3341 Commented Jul 15, 2012 at 3:51
• Thank you Dan. Prop 5.3 indeed answers my second question. So far I can't really tell what difference the fact that the spaces are disconnected makes, but I guess it will become obvious when I try to fill in the details for the proof. Commented Jul 16, 2012 at 11:31
• Not much difference; mainly just notational! Commented Jul 19, 2012 at 17:52

The subject is really way too big (as are so many others of course). I worry a lot about students not in Cambridge or Chicago or Stanford or other places where there are people with folklore at their fingertips. For spectral sequences as a tool, there is a lot to be said for McCleary's guide. Kate Ponto and I just published a book this year, More Concise Algebraic Topology, that may be usable for localizations and completions (just the old-fashioned localize or complete at a set of primes) and that also gives a reasonable start on model categories. Even with that limited scope, the book is much longer than we would like: there were just too many basic details and tools not well enough documented in the literature. There are quite a few other books that go into one or another aspect of the subject (Goerss-Jardine, Neisendorfer, Strom, or, earlier, Whitehead), but it is not to be expected that a single source will cover the ground.

• Even if people with "folklore at their fingertips" are around to give helpful input, most students still need a while and backup by books to digest what they were just told. Maybe it's time to copy the idea the algebraic geometers had with their stack project, i.e. a wiki-like "book" which is not constraint in size. Or simpler, a wiki just with the naked statements ordered by area and references to proofs. Commented Jul 13, 2012 at 14:34
• CAT project (concise AT) might actually be a good name - unless it's already reserved by category people. Commented Jul 13, 2012 at 14:36
• If you happen to be looking for another concise algebraic topology book project, I would like to humbly request a book on the theory of spectra - something which picks up where the final chapter in "A Concise Course in Algebraic Topology" leaves off. I've read the relevant chapter in "Stable Homotopy and Generalized Homology" which seems to be the only available reference for students, and I still feel woefully unprepared to tackle more recent literature on the subject. Commented Jul 13, 2012 at 17:48
• That is actually on my to do' list. I hope to get to it someday. A hopefully student friendly introduction, but with perhaps too much focus on our own approach for your purposes, is `Modern foundations for stable homotopy theory'', by Elmendorf, Kriz, Mandell, and myself (In Handbook of algebraic topology and on my web page). It is easier going than our book. Commented Jul 13, 2012 at 18:54
• I think an AT wiki would be a wonderful thing, especially if it were more readable than nlab! Commented Apr 28, 2013 at 3:42

Re: your first question: As a beginning topologist, I've also been on the lookout for such a text. A book which has looked promising to me is A User's Guide to Algebraic Topology, which can also be found here on Google Books.

• Thank you Aru. I think this book might get too little credit. From the title I expected it to be very elementary, but in fact it does cover some ground. I haven't read it thoroughly but it might be an alternative to Hatcher's book. Commented Jul 16, 2012 at 11:58

You might've (hopefully!) found the answer to your questions by now, but I just came across this post, and in case you're still interested, I find the book 'Algebraic Topology' by Allen Hatcher to be incredibly well-written and self-contained. It's given as a recommended textbook for a lot of elementary algebraic topology courses across many universities - you can find it online at http://www.math.cornell.edu/~hatcher/, together with some other books and links to lecture notes.

All the best,

• I would say that Hatcher's book is much more introductory that what the OP had in mind. Commented Nov 13, 2013 at 21:33

In looking at the analogue of a fibration for squares of spaces you should also look at the "fibrant" condition, which adds to all maps of the square being fibrations the condition that the map from the source of the square to the pullback of the two maps to the target is also a fibration. This condition extended to the case of $n$-cubes is developed in

D. A. Edwards and H. M. Hastings, Čech and Steenrod homotopy theories with applications to geometric topology, Lecture Notes in Math., 542, Springer, Berlin, 1976

and also exploited in

Steiner, Richard; Resolutions of spaces by cubes of fibrations. J. London Math. Soc. (2) 34 (1986), no. 1, 169–176.

Re your second question, I don't have it in front of me but I believe you'll find this in Artin and Mazur's book "Etale Homotopy Theory", near the very beginning of the first section.

• Thanks, for you help. I checked the book and unfortunately I didn't find it. However that doesn't necessarily mean that it isn't there. IMO the book is close to impossible to use as a reference since each section is just one big running text. Commented Jul 13, 2012 at 14:20
• I still don't have Artin and Mazur in front of me, but according to Thomason's paper on Etale Cohomology and Algebraic K-Theory, this should be in Section 1.2. Commented Jul 13, 2012 at 15:22

This is very late, but regarding your second question, in the case of fibrations, see Proposition 7.6.1 in Selick's "Introduction to Homotopy Theory". That section contains a useful discussion of exact sequences in homotopy theory. For the analogous statement for cofibrations to hold, additional assumptions such as simply connectedness are needed.