All Questions
Tagged with galois-theory at.algebraic-topology
16 questions
3
votes
0
answers
158
views
What is the meaning of local inertia conjugation property?
In Hatcher, Allen; Lochak, Pierre; Schneps, Leila, On the Teichmüller tower of mapping class groups, J. Reine Angew. Math. 521, 1-24 (2000). ZBL0953.20030., we have:
Abstract. Let $\widehat{G T}^{1}$ ...
- 119
3
votes
2
answers
336
views
English literature close to "Algébre et Théories Galoisiennes" by Régine and Adrien Douady
I'm currently working on my undergraduate dissertation. I'm working on covering sapces of Riemann surfaces so my supervisor asked me to read the book I mention in the title: "Algébre et Théories ...
- 133
10
votes
2
answers
496
views
Copies of topological fundamental groups inside etale fundamental groups given by different embeddings of your field into $\mathbb{C}$
Let $X$ be a smooth curve over a number field $K$ (not necessarily proper). Fix an algebraic closure $\overline{K}$ of $K$.
Let $i,i' : \overline{K}\hookrightarrow\mathbb{C}$ be two abstract ...
- 10.7k
6
votes
1
answer
520
views
Correspondence between coverings and field extensions
I am self reading from Groups as Galois Group by Helmut Volklein
There is a result on page 94(section 5.4)
Let $G$ be a finite group. Let $P\subset P^{1}$ finite and $q\in P^{1}\P$. There is a ...
- 591
9
votes
0
answers
699
views
Motivic Galois theory and Betti realizations?
Why Motivic Galois groups are defined with Betti realizations? (In fact Absolute Galois groups can be defined in this way (with Betti realizations), why they are so related?).
- 99
14
votes
1
answer
1k
views
Concept of "Rigidity" in mathematics
I am reading from the book "Topics in Galois Theory" by Serre.
I came across the word "Rigidity". I am not able to understand this concept.
If I am not wrong, This term was first used by Thompson, ...
- 591
13
votes
0
answers
287
views
Actions of $\mathbb Z/2\mathbb Z$ on algebraically closed fields and even-dimensional spheres and parallel between Galois theory and covering theory
It is well known that there is a parallel between Galois theory and covering theory. So I wonder whether there is a deep similarity between the following two facts:
Artin-Schreier theorem. The only ...
- 1,980
-2
votes
1
answer
271
views
Any galois covering of $P^{1}$ over rationals are of the form $\mathbb{P}^1_L\to\mathbb{P}^1_\mathbb{Q}$
I recently came across the following statement,
The Galois coverings of $\mathbb{P}^1_\mathbb{Q}$ are all of the form
$$\mathbb{P}^1_L\to\mathbb{P}^1_\mathbb{Q}$$ where $L$ is a number field.
How ...
- 591
-1
votes
1
answer
261
views
Question related to Galois covering of Projective line over rational numbers
Suppose that we have Galois covering $$X\longrightarrow P^{1}_{\mathbb{Q}}$$ defined over the rationals which is cyclic, in the sense that the Galois group associated to the covering is a cyclic group....
- 591
4
votes
0
answers
295
views
Galois categories and the connected components functor
In stacks 0BMQ, a Galois category is defined to be a functor $F:\mathsf C\longrightarrow \mathsf{FinSet}$ such that $\mathsf C$ is finitely bicomplete, every object ...
- 10.5k
0
votes
1
answer
232
views
Inverse Galois Problem ; Galois group of some branched cover of $P^{1}$ defined over $\mathbb{Q}$
I was trying to read a paper on Inverse Galois problem . I understands what the inverse Galois problem is. It asks if every finite group is the Galois group of some extension of the rationals.
The ...
- 591
5
votes
1
answer
314
views
abelian and nonabelian parts of Aut($\widehat{F_2}$)
Let $F$ be the free profinite group on two generators. Let $\text{IA}(F) := \ker\left(\text{Aut}(F)\rightarrow GL_2(\widehat{\mathbb{Z}})\right)$, the group of "IA automorphisms" of $F$. (I'm also ...
- 10.7k
7
votes
1
answer
640
views
Coverings/Cech cohomology of totally disconnected spaces
For any topological space $X$ we have a natural functor
$\text{Cov}_X \rightarrow \text{Fun}(\pi_1(X),\text{Set})$
from the category of coverings of $X$ to the category of functors $\pi_1(X) \...
- 1,704
0
votes
1
answer
197
views
Hurwitz's construction of simple covers
What is commonly meant by Hurwitz's construction of simple covers?
- 3,779
141
votes
0
answers
13k
views
Grothendieck-Teichmüller conjecture
(1) In "Esquisse d'un programme", Grothendieck conjectures
Grothendieck-Teichmüller conjecture: the morphism
$$
G_{\mathbb{Q}} \longrightarrow Aut(\widehat{T})
$$
is an isomorphism.
Here $...
- 7,527
2
votes
1
answer
502
views
Hurewicz theorem related to Galois group (or Tannakian categories)?
Is there a proof of the Hurewicz theorem $\pi_1(X)^{ab} = H_1(X, \mathbf Z)$ ($X$ a connected topological space) expressing $\pi_1(X)$ as the "Galois" group of $X$, i.e., group of deck transformations ...
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