TITLE: Applications of Galois theory for topology
QUESTION [12 upvotes]: Is there any applications of Galois theory in topology? I already have learned Galois theory, and applied it in algebra. Can I get solution of some big problem about topology using Galois theory? Thanks for your help!

REPLY [3 votes]: The  paper 
R. Brown and G. Janelidze,  `A new homotopy double groupoid of a map
of spaces', Applied Categorical Structures 12 (2004) 63-80.
preprint here shows how Janelidze's generalised Galois theory implies the existence of a strict homotopy double groupoid of a map of spaces, generalising previous constructions; the proof is also given directly. 
Edit: @sudiosus: fixed the link -thanks!  I can't help too much with the question about Grothendieck and Galois theory, except to say that the book by Borceux and Janelidze on "Galois theories" does relate the extension to  that from fields to rings, and the referenced paper uses an even further extension, described in the book.