The fundamental-group tag has no wiki summary.
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Fundamental group of row of spheres [migrated]
The fundamental group of $S^1$ is $\mathbb Z$. Let's also call that space $P_1$. Then we'll build $P_n$ for $n > 1$ by taking $P_{n-1}$ and adjoining a circle to it with the condition that it must ...
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local systems with cyclic monodromy
In their book "Lectures on vanishing theorems", Esnault and Viehweg used finite cyclic covering of varieties constructed as follows:
Let $X$ be a smooth projective variety over some field $k$ of ...
2
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1answer
158 views
fundamental group and torus action
Let $T$ be the complex torus acting on a complex connected algebraic variety $X$
and let $p \colon X\rightarrow Y$ be a good quotient for this action.
For any $y\in Y$ we have a sequence $p^{-1}(y) ...
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1answer
202 views
Computing the fundamental group of a flag variety
Let $G$ be a compact and connected and simply connected Lie group and $\mathfrak{g}$ be its Lie algebra and $x\in\mathfrak{g}^*$. How can we compute the fundamental group of $G/G_x$ where $G_x$ is ...
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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 ...
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1answer
176 views
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 ...
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1answer
103 views
A formula for isotropy group $\pi_1(G_a)$
Let $G$ be a compact Lie group and $T$ be its maximal tours, and $a\in \mathfrak{g}^*$. and $G_a$ be the isotropy group of $G$ then $T\subset G_a$ and we know that $\pi_1(T)=\mathbb{Z}^n$. My ...
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Nori fundamental group and etale fundamental group in positive characteristic
Let $X$ be a smooth projective surface over an algebraically closed field of char $p > 0$. Suppose that $\pi_{1}^{et}(X) = \{1\}$. Can Nori fundamental group scheme of $X$ be non-trivial?
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About maps inducing bijections on homotopy classes
Let us assume that $f:X \to Y$ is a map of connected CW complexes, having the following property: if $K$ is a finite CW complex, then the induced map $f_{\ast}:[K,X] \to [K,Y]$ on \emph{free} homotopy ...
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1answer
138 views
Find a simple closed curve in $S$ which represents a commutator in $\pi_1 S$
I am interested in the following problem : decide if a certain element of the fundamental group can be represented by a simple closed curve. The general case has already been asked and answered on MO ...
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1answer
251 views
Computing fundamental groups of the complement of plane curves
This paper of Zariski contains this statement: If $C$ is a curve in $\mathbb{CP}^2$, and $L$ is a generic line, then the injection $L\setminus C \hookrightarrow \mathbb{CP}^2\setminus C$ induces an ...
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2answers
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homotopy exact sequence for the étale fundamental group
I have been trying to understand the homotopy exact sequence for the étale fundamental group which says
$ 1 \rightarrow \pi_1 \bar{X},\bar{x_0})\rightarrow \pi_1 (X,x_0)\rightarrow Gal(k)\rightarrow ...
7
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1answer
322 views
Do all varieties have only finitely many etale covers of fixed degree
I've been wondering about the following "finiteness statement" concerning etale covers for a while.
Let $K$ be a field of characteristic zero, not necessarily algebraically closed. A variety over $K$ ...
7
votes
1answer
340 views
Do elements of the fundamental group give rise to isometries
Let $X$ be a complex algebraic variety, and let $\tilde X\to X$ be its universal cover. Suppose that there exists a Kahler-Einstein metric on $\tilde X$. Note that $\pi_1(X) \subset Aut(\tilde X)$.
...
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3answers
395 views
Smooth projective varieties with infinite abelian fundamental group and finite $\pi_2$
Let $X$ be a smooth projective complex algebraic variety of general type. Suppose that the (topological) fundamental group of $X$ is an infinite abelian group and that $\pi_2(X^{an})$ is finite.
What ...
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2answers
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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) ...
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1answer
166 views
The fundamental group of an $S^1$-quotient
Let $M$ be a compact manifold with an $\mathbb S^1$-action that fixes a point on $M$.
Is it correct that $\pi_1(M/S^1)=\pi_1(M)$?
I believe this is correct and is a corollary of some well-known ...
3
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1answer
168 views
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 ...
4
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1answer
445 views
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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2answers
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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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1answer
312 views
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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1answer
223 views
$P^1$ minus k points
For $k\geq 3$, and $k$ arbitrary points $S=( z_1,\cdots,z_k ) \in \mathbb{P}^1$, we can write
$$ P^1 \setminus S \cong \mathbb{H}/G $$
where $\mathbb{H}$ is the upper-half plane and $G\subset ...
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1answer
332 views
When is the class of functions between sets a set?
I'm reading the paper 'The big fundamental group, big Hawaiian earrings and the big free groups'. The authors state that the class of homotopy equivalences of loops in the space he dubs as the big ...
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1answer
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Is $\pi_1(\widetilde{X/G})$ always finite if $\pi_1(X)$ is finite?
Let $X$ be a smooth complex manifold with finite fundamental group. Suppose that a finite group $G$ acts on $X$ and let $\widetilde{X/G}$ be a resolution of singularities. Is $\pi_1(\widetilde{X/G})$ ...
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2answers
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About “de-Rham” and “l-adic” local systems - comparison
Hello,
Suppose that $k$ is an algebraically closed field of char. 0.
Let $X$ be a smooth connected variety over $k$.
Then I have the category $A$ of Regular Singular smooth $D$-modules on $X$ (i.e. ...
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1answer
303 views
fundamental group of a compact manifold
why fundamental group of of compact manifold is finitely presented
5
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1answer
227 views
Conjugation of homogeneous spaces
Let $X$ be a smooth irreducible algebraic variety
over the field of complex numbers ${\mathbb{C}}$.
Let $x\in X({\mathbb{C}})$.
Let $\tau$ be an automorphism of ${\mathbb{C}}$ (not necessarily ...
1
vote
1answer
337 views
Fundamental Group and Etale Cohomology
I encountered the following statement without a reference many times. For a smooth variety $X$ over a perfect field $k$.
$Hom(H^1_{et}(X, \mathbb{Z}/n), \mathbb{Z}/n) \cong \pi^{ab}_1(X)/n$
Is there ...
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2answers
700 views
Profinite groups as étale fundamental groups
Does every profinite group arise as the étale fundamental group of a connected scheme?
Equivalently, does every Galois category arise as the category of finite étale covers of a connected scheme?
...
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1answer
257 views
Abelianized fundamental group of a curve over a finite field
Let $X$ be a smooth, projective, and geometrically connected curve over a finite field $\mathbb{F}_q$ and fix a geometric point $\overline{x} : \text{Spec } \overline{\mathbb{F}_q} \to X$. Then there ...
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1answer
270 views
Grothendieck's section conjecture and base change: restricting sections
Let $X$ be a smooth projective geometrically connected curve over $\mathbf{Q}$ of genus at least two. Fix an algebraic closure $\overline{\mathbf{Q}}$ of $\mathbf{Q}$ and let $G_{\mathbf{Q}}$ be the ...
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1answer
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local fundamental group of elliptic singularities
Is the local fundamental group of an elliptic singularity virtually solvable ? Here (the terminology is sometimes divergent) an elliptic singularity is a (germ of) normal surface $(X,x)$ such that $X$ ...
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2answers
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Is the fundamental group functor a left-adjoint?
Theorem 1B.9 in Hatcher's Algebraic Topology says that for a (pointed) connected CW complex $X$ and group $G$, there is a bijection $\text{Hom}(\pi_1(X), G) \cong [X,K(G,1)]$, where $\pi_1(X)$ is the ...
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1answer
682 views
Is every ''group-completion'' map an acyclic map?
I start with a longer discussion which will result in a precise version of the question. A am puzzled about an issue with the
Quillen plus construction. I have seen outstanding experts being confused ...
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1answer
518 views
Manifolds with prescribed fundamental group and finitely many trivial homotopy groups
Fix $G$, a finitely generated presented group.
It is known that for every $k > 3$ there is a closed $k$-manifold whose fundamental group is $G$. Similarly, there is a topological space with ...
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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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0answers
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discretifications of the fundamental group functor
Grothendieck calls a "discretification" of a profinite group $\hat G$, a
discrete group $G$ whose profinite completion is isomorphic to $\hat G$.
Does Grothendieck also define a notion of the ...
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3answers
869 views
$\pi_1$ Sequence of Topological Groups
Consider a connected topological group $G$ (not necessarily Lie). You have some maps $G\times G\to G$, such as projection to either summand, or multiplication $(g,h)\mapsto gh$. Now let's look at a ...
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3answers
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When is a three-manifold deck transformation group solvable?
Suppose that $\pi:Y \to Y'$ is a regular covering of closed, connected, orientable three-manifolds and let $G$ be the deck transformation group. Furthermore, suppose that $Y$ is a rational homology ...
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Codimension Two Embeddings in Goodwillie-Weiss Manifold Calculus, and the Difficulty of Fundamental Groups
In manifold calculus, there are various analyticity estimates which run into trouble for codimension two embeddings. For instance, the functor $\operatorname{Emb}(M,N)$ is analytic in $M$ if $\dim M ...
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6answers
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connected compact semisimple lie group finite fundamental group
I was told that the fundamental group of a connected, compact, semisimple Lie group is finite, with the outline of a possible way to prove this fact. Is there any source however that fleshes this out ...
3
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1answer
341 views
A question about the Tannakian etale fundamental group of a curve
Let $X$ be a smooth connected quasi-projective curve over $\mathbf{Q}$. Let $U$ be the pro-unipotent etale fundamental group of $X$ over $\mathbf{Q}_p$.
$U^1 = U$ and let $U^n =[U,U^{n-1}]$.
Let ...
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2answers
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Homology of Covering Spaces
Let $A$ be a subgroup of a group $G$. Then since $A$ is a subgroup of the fundamental group $\pi_1(K(G,1))=G$, there is a covering space $p\colon Y\to K(G,1)$ with $p_*(\pi_1(Y))=A$. So the homology ...
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1answer
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fundamental groups of smooth projective variety.
Is there a discrete group G which is the fundamental group of a compact Kahler
manifold but which is not the fundamental group of any smooth projective complex algebraic variety?
It is known that ...
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3answers
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Can we define homotopy groups using Tannakian categories
This is another vague question. Hope you guys don't mind.
Let $T$ be a Tannakian category. For any fibre functor $F$ on $T$ we define the fundamental group of $T$ at $F$, denoted by $\pi_1(T,F)$, to ...
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2answers
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Number of connected components of the Hurwitz space $H_n^o$ and subgroups of the fundamental group
A cover (of $\mathbf{P}^1_{\overline{\mathbf{Q}}}$) is a finite morphism $X\to \mathbf{P}^1_{\overline{\mathbf{Q}}}$, where $X$ is a smooth projective connected curve over $\overline{\mathbf{Q}}$. ...
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2answers
565 views
Are acyclic subcomplexes of finite contractible 2-complexes contractible?
Let $Y$ be a contractible finite simplicial 2-complex.
Let $X$ be an acyclic subcomplex of $Y$ (i.e. $X$ connected, $H_1(X)=0$, $H_2(X)=0$).
Is $X$ contractible? (Equivalently, is $\pi_1(X)$ ...
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3answers
842 views
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 > G_1 > ... > ...
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3answers
608 views
3-manifolds with solvable fundamental group
Is there a nice reference for the classification of closed 3-manifolds with solvable (nilpotent, abelian, etc.) fundamental group, assuming the Geometrization Conjecture?
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3answers
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Tutte polynomials of appropriate Cayley graphs
I was quite intrigued by Tutte polynomials in a recent talk I had been to. It was introduced as a polynomial associated to a undirected finite graph. For a graph $G=(V,E)$ we form the polynomial
...
