12

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!

  • Sure. Google "Khovanskii Topological Galois theory". –  Feb 10 '14 at 11:15
  • 3
    http://math.stackexchange.com/questions/35503/a-galois-group-is-a-fundamental-group – Ferra Feb 10 '14 at 11:18
  • I would also reccomend, depending on your background, to look at Galois theory for Schemes by Lenstra. Szamuely is one of my favorite math books, and is a must read, but Lenstra very explicitly outlines the categorical nonesense underlying all of these manifestations of "Galoisness". – Alex Youcis Feb 10 '14 at 11:30
  • 2
    @user127249: This is a very nice example, but the one which goes in the opposite direction (from topology to algebra). – Moishe Kohan Feb 15 '14 at 13:52

3 Answers3

6

I'm not sure if this is what you're looking for, but there is a Galois correspondence for covering spaces and deck transformations that is analogous to the correspondence between intermediate fields and field extension automorphisms of a Galois extension.

Viktor Vaughn
  • 19,278
3

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.

Ronnie Brown
  • 15,309
  • First, could you fix the link, since currently it points to some "official disclaimer" web-page with no math content. Secondly and more importantly, I have the same problem with this example as with other examples where covering theory or its generalization is treated as a "Galois Theory" (I include the one by Grothendieck here too). This is all fine and well, but any direct connection to the number theory is missing here. Do you know any closer examples than Sullivan's notes in my answer (and subsequent papers by Deligne and Sullivan and a book by Eric Friedlander)? – Moishe Kohan Feb 15 '14 at 13:48
2

Take a look at the book "Geometric Topology Localization, Periodicity, and Galois Symmetry" by Dennis Sullivan. The book (as almost everything that Sullivan wrote) is hard to read, but you can just browse it to get an idea of what it is about.

Moishe Kohan
  • 97,719