Questions tagged [fundamental-group]
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32 questions
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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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Geometric interpretation of the lower central series for the fundamental group?
For any group $G$ we can form the lower central series of normal subgroups by taking $G_0 = G$, $G_1 = [G,G]$, $G_{i+1} = [G,G_i]$. We can check this gives a normal chain
$$G_0 \ge G_1 \ge ... \ge G_i ...
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Reference request on birational invariance of Chow group of zero cycles of degree zero
Let $CH_0(X)^0$ denote the group of zero cycles of degree zero modulo rational equivalence.
I am looking for a reference for the following fact:
If $X$ and $Y$ are smooth and projective varieties ...
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Fundamental group as topological group
Background
Let $(X,x)$ be a pointed topological space. Then the fundamental group $\pi_1(X,x)$ becomes a topological space: Endow the set of maps $S^1 \to X$ with the compact-open topology, endow the ...
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Is there a manifold with fundamental group $\mathbb{Q}$?
It is known that the fundamental group of a locally path connected, path connected compact metric space is finitely presented or uncountable. Furthermore the fundamental group of every manifold is ...
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Fundamental group of Lie groups
Let $T$ be a torus $V/\Gamma$, $\gamma$ a loop on $T$ based at the origin. Then it is easy to see that $$2 \gamma = \gamma \ast \gamma \in \pi_1(T).$$
Here $2 \gamma$ is obtained by rescaling $\gamma$...
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What properties make $[0,1]$ a good candidate for defining fundamental groups?
The title essentially says it all. Consider the category $\mathfrak{Top}_2$ of triples $(J,e_0,e_1)$ where $J$ is a topological space, and $e_i \in J$. There is an obvious generalization of the ...
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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 ...
- 2,806
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4
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Is $\mathbb{R}^3 \setminus \mathbb{Q}^3$ simply connected?
Similarly is the complement of any countable set in $\mathbb R^3$ simply connected?
Reading around I found plenty of articles discussing the path connectedness $\mathbb R^2 \setminus \mathbb Q^2$ and ...
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Nonabelian topological fundamental group of a conjugate variety
Let $X$ be a pointed algebraic variety over the field of complex numbers $\mathbb{C}$.
Let $\pi_1^{\rm top}(X)$ and $\pi_1^{\mathrm{\acute{e}t}}(X)$ denote the topological and the étale fundamental ...
- 14.2k
23
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3
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How bad can $\pi_1$ of a linear group orbit be?
Let $G$ be a simply connected Lie group and $\mathcal O= G(v)=G/G_v$ a $G$-orbit in some finite-dimensional $G$-module $V$. By the homotopy exact sequence, its fundamental group $\Gamma$ is the ...
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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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"Cute" applications of the étale fundamental group
When I was an undergrad student, the first application that was given to me of the construction of the fundamental group was the non-retraction lemma : there is no continuous map from the disk to the ...
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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 ...
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Teaching the fundamental group via everyday examples
This question is a "prequel" to a similar question about homology. Both questions were inspired by seeing a talk, by Tadashi Tokieda, about the interesting physics that appears in toys.
What ...
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Why are we interested in the Fundamental Groupoid of a Space?
The classical version of the van Kampen theorem is concerned about the fundamental group of a based space. In fact, it says that the functor $\pi_1$ preserves certain types of pushouts in $Top_*$.
...
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How should one understand orbifold fundamental groups?
I am studying orbifold fundamental group (or more generally orbifold homotopy groups). In a nutshell, my questions is: what are they intuitively? In what follows I give definitions and more precise ...
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Does anyone know a basepoint-free construction of universal covers?
Let $X$ be a real manifold (for simplicity). The standard construction of the universal cover $\varphi: \widetilde{X} \longrightarrow X$ involves fixing a basepoint $p \in X$ and considering homotopy ...
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On a homological finiteness condition
Assumption: $X$ is a connected CW complex, and $H_{\ast}(X;\mathbb{Z})=\bigoplus_{n \geq 0} H_n (X; \mathbb{Z})$ is finitely generated.
Question: does there exist a finite CW complex $Y$ and a map $f:...
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The fundamental group of a closed surface without classification of surfaces?
The fundamental group of a closed oriented surface of genus $g$ has the well-known presentation
$$
\langle x_1,\ldots, x_g,y_1,\ldots ,y_g\vert \prod_{i=1}^{g} [x_i,y_i]\rangle.
$$
The proof I know ...
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answer
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What is the wild fundamental group?
In the abstract of
Singularités irrégulières Correspondance et documents
Pierre Deligne, Bernard Malgrange, Jean-Pierre Ramis
Documents mathématiques 5 (2007), xii+188 pages (link)
there is a ...
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locally constant constructible sheaves and finite etale coverings
Maybe it is well known to experts or maybe it is just a stupid idea, but I will ask any way.
We know that if $X$ is a topological space, then there is an equivalence of categories between the ...
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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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Under what conditions is the induced map of etale fundamental groups surjective?
Let $f:X \to Y$ be a morphism of schemes. I am interested in sufficient conditions on $f$ which would ensure that the induced map $\pi_1^{et}(X) \to \pi_1^{et}(Y)$ of etale fundamental groups is ...
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Fundamental group of an hyperbolic $4$-manifold
Good afternoon everyone,
I have a very general question about hyperbolic manifolds and their fundamental groups in high dimension (at least $4$). If the theory of surfaces and $3$-manifolds provide ...
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3
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Reference needed: Isomorphism on pi_1 and homology gives weak equivalence
Let $f : X \to Y$ be a map between a connected space $X$ and a space $Y$. If $\pi(f) : \pi_1(X) \to \pi_1(Y)$ is an isomorphism, and $H_n(f) : H_n(X, G) \to H_n(Y, G)$ is an isomorphism for all $n \ge ...
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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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Algebraic numbers and the complex projective line minus three points
Deligne's monograph “Le Groupe Fondamental de la Droite Projective Moins Trois Points” begins by remarking that when X is the projective line over the complex numbers, minus three points: "every ...
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The fundamental group of a $3$-manifold with a boundary of genus $>0$
Let $M$ be an orientable $3$-manifold with connected boundary $\Sigma_g$, a surface of genus $g>0$.
I would like to find a reference to the following two statements.
1) $\pi_1(M)\ne 0$.
2) $\...
- 14.3k
4
votes
1
answer
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Structure of fundamental groups arising from smooth projective morphisms
Let $f:X\to B$ be a smooth projective morphism of complex algebraic varieties.
If $f$ is of relative dimension zero, i.e., $f$ is a finite etale cover, then the image of the topological fundamental ...
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Finding the 2nd homotopy group $\pi_2(G^\mathbb{C}/P)$
Let $G$ be a compact connected and simply connected Lie group and $G^\mathbb{C}$ be the complexification of Lie group (with is diffeomorphic with $G^\mathbb{C}\cong T^*G$) then I am looking for ...
user21574
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Trying to relate the fundamental groupoid to vector bundles
Fix a topological space $X$. Now consider a functor from the fundamental groupoid of $X$ to the category $Vect$. In other words, we assign a vector space to each point of $X$, we allow ourselves to ...
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