TITLE: Grothendieck's Tohoku Paper and Combinatorial Topology
QUESTION [21 upvotes]: I've read some discussions of Grothendieck's famous Tohoku Paper, and I understand that one reason it was a landmark paper was that it introduced abelian categories and gave us sheaf cohomology as a certain type of derived functor. However, I've heard from various sources (Manin's Homological Algebra, my prof, and the 2 part AMS Notices articles) that one of the famous aspects of this paper is that it "reinterpreted" the basics of combinatorial topology.  Does anybody know what this means?  Slightly more specific: how was combinatorial topology understood at that time and how did the Tohoku paper force a reinterpretation of the conventional concepts? I know this question might be a bit ill posed, but I'd like to know if somebody understands what this means and can explain it at a first year graduate student level.
Thanks,
Ben

REPLY [19 votes]: One important piece of information in the Tohoku paper is that for X a finite CW complex and G a finite group acting on X, there is an isomorphism $H^* (X/G; K) \simeq H^* (X; K)^G$, where $K$ is a field of characteristic zero.  I don't know of a way to see this without sheaf cohomology.  Note that the action need not be free.  
(MacDonald's paper on the cohomology of symmetric products references this result to the Tohoku paper, specifically Theorem 5.3.1 and the Corollary to Proposition 5.2.3.)
This is an example of how reformulating algebraic topological concepts sheaf-theoretically can be very helpful, and maybe that's the sense in which the paper "reinterpreted combinatorial topology."