TITLE: Algebra for algebraic topology
QUESTION [5 upvotes]: 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Algebraic Topology plus Eilenberg-Steenrod's axioms (not included in Hatcher's book)  plus spectral sequences (in unpublished notes of Hatcher).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.

REPLY [3 votes]: 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.