TITLE: What is the possible usefulness of étale topology and cohomology apart from the resolution of the Weil conjecture ?
QUESTION [5 upvotes]: 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 ?

REPLY [2 votes]: 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).