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My research is in analysis, but it moved to the area that requires algebraic topology. I have some working knowledge in that area, but I always feel that I am on a shaky ground and I need to go back and study algebraic topology again. However, that would also require refreshing my knowledge in algebra. My question is:

What is a good reference from which I could learn algebra necessary for studying algebraic topology at the level of Hatcher's Algebraic Topology plus Eilenberg-Steenrod's axioms (not included in Hatcher's book) plus spectral sequences (in unpublished notes of Hatcher).

I would love to find an elementary reference that would cover all necessary algebraic tools (including homological algebra) on no more than 100$\pm\varepsilon$ pages.

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    $\begingroup$ That depends very much on what parts of algebraic topology you want to use and for what purpose. Further details would be helpful. $\endgroup$ – Neil Strickland Dec 29 '18 at 21:46
  • $\begingroup$ @NeilStrickland I added more details to my question. $\endgroup$ – Piotr Hajlasz Dec 29 '18 at 22:03
  • $\begingroup$ Serge Lang's Algebra covers what you need, but it's more than 100 pages. Dummit and Foot is a nice undergraduate algebra book that covers essentially all you need, although it does not really develop homological algebra -- but you don't really need to know any homological algebra to study from Hatcher's book. Dummit and Foot is also over 100 pages. I imagine there is something that satisfies all your criteria. The undergraduate books all are a little longwinded compared to what you are looking for. $\endgroup$ – Ryan Budney Dec 29 '18 at 22:10
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    $\begingroup$ @RyanBudney Dummit and Foote: 932 pages, Lang: 914 pages. That is precisely what I would like to avoid. I actually do not like the book by Dummit and Foote: a lot of unnecessary words, like in most of the undergraduate texts. $\endgroup$ – Piotr Hajlasz Dec 29 '18 at 22:28
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    $\begingroup$ amazon.com/Algebra-Chapter-Graduate-Studies-Mathematics/dp/… is again long, but covers much more ground than Dummit & Foote (and you can safely skip the irrelevant to you chapters, it's much nicer structured than D and F IMHO) $\endgroup$ – Dima Pasechnik Dec 30 '18 at 3:55
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If you need a concise but very clear book which covers a lot of Algebraic Topology and just the necessary algebra (spectral sequences as well) I think that Differential Forms in Algebraic Topology- Bott & Tu is the book you are looking for.

Edit: It seems that Bredon-Topology and Geometry is closer to that you are looking for.

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    $\begingroup$ I know this book, but it does not really provides a good introduction to algebra. Moreover, the deRham approach to topology gives only real coefficients and neglects the torsion part. $\endgroup$ – Piotr Hajlasz Dec 29 '18 at 23:06
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    $\begingroup$ Ah! Probably I found the right textbook! Try to take a look at Topology and geometry-Bredon :) It covers the necessary algebra along the way and begins with the basic topics of Algebraic Topology. $\endgroup$ – Vincenzo Zaccaro Dec 29 '18 at 23:11
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    $\begingroup$ I need to have a closer look at the book by Bredon, but it seems it might be an approach that I like, geometric and quite diverse. $\endgroup$ – Piotr Hajlasz Jan 6 '19 at 2:44
  • $\begingroup$ It also introduces Eilenberg-Steenrod's axioms for a general Homology Theory. Anyway...I'm happy to help :) $\endgroup$ – Vincenzo Zaccaro Jan 6 '19 at 2:49
  • $\begingroup$ I edited my answer :) $\endgroup$ – Vincenzo Zaccaro Jan 6 '19 at 2:59
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I think what you need is a book on Homological algebra that discusses some category theory, some homology and group cohomology. You can try

A Course in Homological algebra by Peter Hilton and Urs Stammbach

You can read first two chapters (first chapter talks about modules second chapter talks category theory) and then read 6th chapter on group cohomology. These comes around 130 pages.

There is a 5 page section named "Homological Algebra and Algebraic Topology". This is about application of Homological algebra in Algebraic Topology. You can read this first.

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