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It has so happened that I have come this far knowing nothing on the subject of algebraic topology (as in homology theories of topological spaces and their applications). I've decided to finally read up on that during the summer.

Seemingly, however, the authors of most books for beginners are hesitant to make use of nontrivial homological algebra and category theory, which, if I'm not mistaken, could be used to speed up and at the same time clarify the presentation. I, on the other hand, would dare say to be somewhat familiar with these disciplines. (I'm, to different degrees, acquainted with derived functors, spectral sequences, derived categories as well as sheaf cohomology and Lie algebra/group cohomology.)

Thus, what I'm looking for is an introduction to algebraic topology the author of which readily employs the above concepts when appropriate.

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My view is that it's better at the start to work as concretely as possible so you can see clearly (geometrically) how the algebraic objects really do describe interesting topological properties. At the beginning, the abstract machinery might obscure more than enlighten, even if you're comfortable with the machinery. – Deane Yang Jul 8 '13 at 13:34
Spanier's? Switzer's? Is Hatcher's too basic for you? – Fernando Muro Jul 8 '13 at 13:56
I'll write you such a textbook here: ``there's an acyclic resolution of the constant sheaf that some people like; the i-th term in this complex is the sheaf which associates to an open set U the space of functions on the set of maps of an i simplex into U.'' – Vivek Shende Jul 8 '13 at 13:59
@vivekshende who on earth are you talking to? – Vidit Nanda Jul 8 '13 at 14:11
It's not clear to me how a textbook would benefit from bringing in these perspectives. If you have background in sheaves and derived categories, that would perhaps help you digest any of the standard textbooks a little quicker. But I'm not seeing how exposition of basic algebraic topology would be improved using these tools. – Ryan Budney Jul 12 '13 at 1:39

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.

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The link for Peter May's book seems not to work. – Dietrich Burde Jul 8 '13 at 13:45
I second the recommendation of Edelsbrunner and Harer. Very nice read. – Steve Huntsman Jul 8 '13 at 13:51
Thanks, @DietrichBurde it has been fixed – Vidit Nanda Jul 8 '13 at 13:52
@danielmoskovich you can start reading May from Chapter 12, completely bypassing the homotopy theory. But of course, you shouldn't! – Vidit Nanda Jul 8 '13 at 14:14
Thanks to several people. I've checked the link to Concise and it seems to work fine --- There is a sequel now on line too, More (or less) Concise – Peter May Jul 8 '13 at 20:54

There's a great book called Lecture Notes in Algebraic Topology by Davis and Kirk which I highly recommend for advanced beginners, especially those who like the categorical viewpoint and homological algebra. I think the treatment in Spanier is a bit outdated. Davis and Kirk is written with an eye towards what you might learn next, e.g. model categories. By the way, there's a pdf of it available for free here.

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A revised preliminary second edition is available for free here: I highly recommend using this version instead. – Henry T. Horton Jul 8 '13 at 20:53
In the revised edition much of the text is highlighted with all sorts of colours. Should they be ignored? – Igor Makhlin Jul 12 '13 at 19:15
The code doesn't seem to be indicated but I think it's something like green=comments by the authors for future revisions, blue=revised things, red=things that need revision? In it's current condition, it doesn't seem all that readable to me. There is also an online errata for the original version: – Justin Young Jul 17 '13 at 13:58

My sense is you haven't read Allen Hatcher's book closely enough. I certainly need to go through it.

Jacob Lurie had nice Geometric Topology course a few years back, if you like that style. Here's an intriguiging sounding course on Chromatic Homotopy Theory

Akhil Matthew took notes on a course by Michael Hopkins.

You may wish to delve into the literature more directly. Have you looked into Twisted K-Theory ?

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A recent book is tom Dieck's "Algebraic Topology", which is precisely written and quite comprehensive. But I've only skimmed it, so I'd be interested in more expert opinions.

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I've worked through parts of this book and am a big fan. I would definitely recommend it to the OP. The approach is also more modern than Spanier. – Adeel Khan Jul 8 '13 at 20:04
But the OP would not like it, since tom Dieck carefully avoids spectral sequences. – Johannes Ebert Jul 9 '13 at 8:07

You might find Jeff Strom's new book attractive. Here is a review.

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I think you're describing Spanier.

Everyone I know who has seriously studied from Spanier swears by it- it's an absolute classic. The approach is exactly as you describe- algebraic topology for grown-ups. The treatment of homological algebra in it is extremely nice, and quite sophisticated.

A second, quite brilliant book along the same lines is Rotman. It's more geometric than Spanier, for those who like such things, and find it easier to read (although that's a matter of taste of course). Again, the treatment is unembarrassed to employ nontrivial homological algebra and category theory, in a good way.

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Wow, Spanier sounds tempting! All in all, there seems to be a lot more to choose from, than I could imagine. Which has an upside and a downside =) – Igor Makhlin Jul 8 '13 at 14:14
From a purely typographical point of view, I find Spanier pretty brutal to read. – Todd Trimble Jul 8 '13 at 15:54
I read Spanier when I was a graduate student and learned a lot from it, but it is pretty old-fashioned (and has some eccentricities; for instance, the chapter on the fundamental group only discusses Seifert-van Kampen in an exercise, and then only for simplicial complexes). – Andy Putman Jul 8 '13 at 23:01
Spanier isn't for everyone; but I think it might be exactly what the OP is looking for. And everyone who I know who took it seriously seems to have gotten a tremendous amount out of it... kind of like Hatshorne or EGA in Alg. Geom. – Daniel Moskovich Jul 9 '13 at 3:54
@ToddTrimble If copyright were nonexistent, it might be wonderful to re-typeset Spanier. Maybe throw in some comments to highlight aspects where the emphasis has shifted with time. Spanier is worth it! – Daniel Moskovich Jul 9 '13 at 3:56

My suggestion:

Algebraic Topology: Homotopy and Homology - by Robert M. Switzer

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There is also an advanced book in homotopy theory which uses concepts you mentioned in your question. Simplicial Homotopy Theory, by P.G. Goerss and J.F. Jardine. – Vahid Shirbisheh Jul 8 '13 at 19:37

You have many good suggestions already. Another book which you might enjoy is "Cohomology Operations and Applications in Homotopy Theory" by Mosher and Tangora.

This is not really a beginner's book per-se, as it assumes a basic knowledge of ordinary cohomology from the start. However it has a lot to recommend it, including brevity, affordability and concreteness (the focus is on applications of cohomology theory to calculations of the homotopy groups of spheres). It also seems to meet your criteria in that it gets quickly to the deeper applications of homological algebra and spectral sequences in homotopy theory.

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