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My question is as stated in the title:

What is the possible usefulness of étale topology and cohomology apart from the resolution of the Weil conjecture ?

I am particularly interested to know if it's reasonably possible to deduce results in areas like 3 and 4 manifolds topology or Kähler geometry from this machinery. Has anyone ever tried ?

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In complex geometry you have the usual de Ram and Betti cohomology groups and the analytic topology, so in some respects you don't really "need" étale cohomology. For me, the whole point of étale cohomology is that it exists over general fields (e.g. finite fields). –  Daniel Loughran Mar 27 '12 at 22:39
As for explicit uses outside the Weil conjectures, étale cohomology groups come equipped with an action of the absolute galois group of the field in question, and hence give rise to naturally occuring Galois representations which are of great interest to number theorists –  Daniel Loughran Mar 27 '12 at 22:39
I edited the English a bit. More important, this is a very broad inquiry which allows for multiple possible answers even in the specific areas you mention. In a more algebraic direction, etale cohomology has been an essential tool in the character theory of finite groups of Lie type as pioneered by Deligne and Lusztig in the mid-1970s and exposed in several books as well as many research papers. The subject is definitely useful beyond the Weil conjectures. –  Jim Humphreys Mar 27 '12 at 22:41
Many interesting answers here: mathoverflow.net/questions/6070/etale-cohomology-why-study-it –  Hunter Brooks Mar 28 '12 at 1:33
Oh well. Here imrn.oxfordjournals.org/content/2002/28/1505.short is a trivial bit of self-promotion, since you did ask about manifold topology. –  Minhyong Kim Mar 29 '12 at 13:50

5 Answers 5

The etale topology (ignoring etale cohomology) underlies the theory of algebraic spaces (which in turn is the basis for the "modern" approach to moduli spaces following Artin).

More concretely, the Artin approximation theorem provides a genuine sense in which the etale topology is the algebraic version of working locally for the complex-analytic topology (similar to the way that the implicit function theorem ensures the existence of convergent solutions to systems of analytic equations admitting a formal solution): in the stronger form proved by Popescu (building on Artin's results) it says that a finite system of polynomial equations over an (excellent) local noetherian ring with a solution in the completion admits solutions in a local-etale extension (even arbitrarily close to the solution in the completion).

Application: the Artin-Popescu approximation theorem enables one to "promote" solutions to relative algebro-geometric problems over completions (built via deformation theory, for example) into solutions over an etale neighborhood of a point in the base. More concretely, it implies that if two algebraic varieties over a field are "formally isomorphic" at a rational point (i.e., have isomorphic completed local rings) then they admit a common etale neighborhood. This is especially effective when trying to study algebraic singularities via their formal cousins (i.e., completions). So a framework for treating etale maps "as if" they are local isomorphisms (i.e., the etale topology!), even in the presence of arbtrary singularities, gives a way to pass back and forth between algebraic situations and formal situations.

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$l$-adic cohomology has been used in a crucial way to investigate the representation theory of finite groups of Lie type. Basically, all irreducible characters of such a group are summands of characters that come from the action on certain $l$-adic cohomology groups. In many cases, these cohomlogy groups already realise all the irreducible representations. This is the Deligne-Lusztig theory.

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I actually find the tone of this question a bit off-putting, but I'll give an answer anyway. The whole subject of modularity of Galois representations is predicated on the fact that Galois groups acts on etale cohomology (as in Daniel's comment) and that this action is well-behaved in a certain way and the hope is (Fontaine-Mazur) that all well-behaved l-adic Galois representations come from geometry. Specifically in low-dim'l topology and Kahler geometry, I don't know.

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Quillen: "Some remarks on etale homotopy theory and a conjecture of Adams."

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Not sure that should count, but still: morally there is relation with Milnor's K-groups.

According to Milnor conjecture proved by Voevodsky 2-torsion in Milnor K-groups are the same as Galois cohomology mod Z/2. Etale cohomology is "globaliztion" of the Galois cohomology, and there are certain "globalizations" of Milnor's conjecture. So you get the relation... (As far as I understand Voevodsky extended this to Z/l coefficients also).

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