Questions tagged [fundamental-group]
The fundamental-group tag has no usage guidance.
268 questions
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Can the fundamental group of any manifold be realized as the fund grp of a finite space?
Recently, I was asked to calculate the fundamental group of the space $X= \{a,b,c,d\}$ with open sets generated by $\{ a, c, abc, acd \}$.
Turns out, $\pi_1(X)\cong \mathbb Z$ and in fact, $X$ is the ...
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Computing `$\pi_1 S^1$` using groupoids
I believe it is possible to compute $\pi_1 S^1$ by applying the groupoid version of the Seifert-Van Kampen Theorem (in the version presented in May's Concise Course) to a covering of the circle by ...
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What is π_1(BG) for an arbitrary topological group $G$?
The classifying space $BG=|Nerve(G)|$ of an arbitrary topological group $G$ does not necessarily have the homotopy type of a CW-complex but the fundamental group should still be accessible. What is $\...
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Fundamental group of Lie groups
Let $T$ be a torus $V/\Gamma$, $\gamma$ a loop on $T$ based at the origin. Then it is easy to see that $$2 \gamma = \gamma \ast \gamma \in \pi_1(T).$$
Here $2 \gamma$ is obtained by rescaling $\gamma$...
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Grothendieck's Galois Theory today
I have recently become aware of, and started to study in my free time (abundant in these summer months) Grothendieck's Galois Theory (GGT), as formulated in SGA 1 and later by Grothendieck's ...
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étale fundamental group of projective space
What is the étale fundamental group of projective space over an algebraically closed field?
In char = 0 it is trivial (Lefschetz principle), as well as in dimension 1 (Riemann-Hurwitz).
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Fundamental group as topological group
Background
Let $(X,x)$ be a pointed topological space. Then the fundamental group $\pi_1(X,x)$ becomes a topological space: Endow the set of maps $S^1 \to X$ with the compact-open topology, endow the ...
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Unipotency in realisations of the motivic fundamental group
Deligne, in his 1987 paper on the fundamental group of $\mathbb{P}^1 \setminus \{0,1,\infty\}$ (in "Galois Groups over $\mathbb{Q}$"), defines a system of realisations for a motivic fundamental group. ...
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Fundamental group of the line with the double origin.
In the simplest cases, the fundamental group serves as a measure of the number of 2-dimensional "holes" in a space. It is interesting to know whether they capture the following type of "hole".
This ...
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Functoriality of fundamental group via deck transformations
Problem
I'm trying to understand this with a view towards the etale fundamental group where we can't talk about loops. What I'm missing is how the fundamental group functor should work on morphisms, ...
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Tannaka formalism and the étale fundamental group
For quite a while, I have been wondering if there is a general principle/theory that has
both Tannaka fundamental groups and étale fundamental groups as a special case.
To elaborate: The theory of ...
- 4,450
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2
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Covers of the projective line over Z and arithmetic Grauert-Remmert
This question is the two-dimensional analogue of Etale coverings of certain open subschemes in Spec O_K
There I asked if one could characterize in a way certain covers of $\textrm{Spec} \ O_K$. As ...
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Fundamental groups of topoi
Just yesterday I heard of the notion of a fundamental group of a topos, so I looked it up on the nLab, where the following nice definition is given:
If $T$ is a Grothendieck topos arising as category ...
- 4,450
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Etale coverings of certain open subschemes in Spec O_K
Let $U$ be an open subscheme of $\textrm{Spec} \ \mathbf{Z}$. The complement of $U$ is a divisor $D$ of $\textrm{Spec} \ \mathbf{Z}$.
Q. Can we classify the etale coverings of $U$ of a given degree? ...
- 9,497
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Fundamental groups of noncompact surfaces
I got fantastic answers to my previous question (about modern references for the fact that surfaces can be triangulated), so I thought I'd ask a related question. A basic fact about surface topology ...
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Simply connected quasi-projective varieties in positive characteristic
I am looking for examples of non-projective (quasi-projective) varieties $X$ defined over a field of positive characteristic, which have trivial étale fundamental group.
It is well known that the ...
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Fundamental group of 3-manifold with boundary
Is it true that any finitely presented group can be realized as fundamental group of compact 3-manifold with boundary?
- 45k
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Galois groups vs. fundamental groups
In a recent blog post Terry Tao mentions in passing that:
"Class groups...are arithmetic analogues of the (abelianised) fundamental groups in topology, with Galois groups serving as the analogue ...
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