# Grothendieck's Tohoku Paper and Combinatorial Topology

Hi,

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

• Retagged. "Combinatorial topology" is what people used to call algebraic topology in the first half of the 20th century. – bhwang Mar 21 '10 at 3:54

## 2 Answers

Grothendieck's Tohoku paper was an attempt to set the foundations of algebraic topology on a uniform basis, essentially to describe a setting where one can do homological algebra in a way that makes sense. He did this by using the concept of abelian categories. Perhaps a better question to ask yourself is "Why are abelian categories a good idea?" In answering your question, I will do some major handwaving and sacrifice some rigor for the sake of clarity and brevity, but will try to place the Tohoku paper in context.

At the time, the state-of-the-art in homological algebra was relatively primitive. Cartan and Eilenberg had only defined functors over modules. There were some clear parallels with sheaf cohomology that could not be mere coincidence, and there was a lot of evidence that their techniques worked in more general cases. However, in order to generalize the methodology from modules, we needed the category in question to have some notion of an exact sequence. This is a lot trickier than it might seem. There were many solid attempts to do so, and the Tohoku paper was a giant step forward in the right direction.

In a nutshell, Grothendieck was motivated by the idea that $Sh(X)$, the category of sheaves of abelian groups on a topological space $X$ was an abelian category with enough injectives, so sheaf cohomology could be defined as a right-derived functor of the functor of global sections. Running with this concept, he set up his famous axioms for what an abelian category might satisfy.

Using the framework given by these axioms, Grothendieck was able to generalize Cartan and Eilenberg's techniques on derived functors, introducing ideas like $\delta$-functors and $T$-acyclic objects in the process. He also introduces an important computational tool, what is now often called the Grothendieck spectral sequence. This turns out to generalize many of the then-known spectral sequences, providing indisputable evidence that abelian categories are the "right" setting in which one can do homological algebra.

However, even with this powerful new context, there were many components missing. For instance, one couldn't even chase diagrams in general abelian categories using the techniques from Tohoku in and of itself, because it did not establish that the objects that you wanted to chase even existed. It wasn't until we had results like the Freyd-Mitchell embedding theorem that useful techniques like diagram chasing in abelian categories became well-defined. Henceforth, one had a relatively mature theory of homological algebra in the context of abelian categories, successfully generalizing the previous methods in homological algebra. In other words, we have "re-interpreted the basics of [algebraic] topology" by allowing ourselves to work with the more general concept of abelian categories.

• IMO saying this is a reinterpretation of the basics of algebraic topology is an overstatement. It's more of an elaboration of homological algebra than a reinterpretation of algebraic topology. – Ryan Budney Mar 21 '10 at 5:40
• You're definitely right. It's titled "Sur quelque points d'algèbre homologique" for a good reason. However, I do not personally know of any other way that the question could be answered/interpreted in a way that makes sense. Maybe an algebraic topologist could give a better answer? – bhwang Mar 21 '10 at 16:40
• I sometimes think of myself as one. :) – Ryan Budney Mar 21 '10 at 22:11
• Yes, this reads like an answer to a similar but not identical question "How did Grothendieck's Tohoku paper reinterpret the basics of homological algebra?" I would say that relevance for algebraic topology is hidden in your 3rd paragraph: before Grothendieck, homology and cohomology were viewed as functions of a topological space (the "non-abelian argument" in Gelfand-Manin's parlance), hence the Eilenberg-Steenrod axioms, whereas Grothendieck retooled them as functions of the sheaf (the "abelian argument", ibid) and opened the door to the methods of homological algebra. – Victor Protsak Jun 30 '10 at 7:14

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."

• One can see this by considering the map $X \times_G EG \to X/G$ from the Borel construction, filtering both sides by skeleta of $X$, and calculate directly that it is true for wedges of spheres (for which it is enough to treat the case where $G$ acts transitively on the spheres: then the actual quotient is a sphere, and the homotopy quotient is a sphere times the classifying space of the stabiliser, a finite group). – Oscar Randal-Williams Jun 30 '10 at 9:24
• Interesting. I've never seen that before. Is it written down anywhere? I happen to be using this fact in a paper at the moment, which is how I stumbled across this question. – Dan Ramras Jun 30 '10 at 14:29
• @Dam. It's been a while since you posted this answer, but I read it right now and I'm interested: which result is this in the Tohoku paper? – Agustí Roig Sep 17 '10 at 8:27
• Agusti, I've edited to include this information. – Dan Ramras Sep 17 '10 at 15:54
• @jdc The result in Hatcher only covers the case in which the action of the finite group $G$ is free (which in that case is equivalent to the quotient map being a covering space projection). The result I'm quoting doesn't assume the action is free. – Dan Ramras Feb 20 '17 at 3:03