The fundamental-group tag has no usage guidance.

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121 views

### $p$-adic representations of the fundamental group of a smooth proper curve over a finite field

This question is very general. Let $C$ be a smooth and proper curve over a finite field ${\bf F}_p$. Are there any general results or conjectures on continuous non abelian representations
$$
\pi_1(C)\...

**4**

votes

**2**answers

224 views

### 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 ...

**4**

votes

**0**answers

64 views

### Nonabelian Hodge structure for noncompact curves and Hodge structure on the fundamental group

Nonabelian Hodge theory, introduced by C. Simpson and others, may be interpreted as a description of the (real) Hodge structure on the fundamental group (say, of a compact curve) in terms of some ...

**3**

votes

**0**answers

76 views

### Do character sheaves split over the Lang isogeny?

Let $G$ be a smooth commutative connected algebraic group over a finite field $\mathbb{F}_q$. For my purposes a character sheaf on G is a rank one $\ell$-adic local system $\mathcal{L}$ on $G$ ...

**1**

vote

**0**answers

40 views

### Restriction of irreducible integrable connections

Let $X$ be a complex analytic manifold and let $\mathcal{M}$ be an integrable connection on $X$, i.e. a $\mathcal{D}_X$-module which happens to be coherent as an $\mathcal{O}_X$-module. If $U$ is any ...

**8**

votes

**1**answer

404 views

### 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 ...

**4**

votes

**1**answer

243 views

### What is kernel $\phi:G\rightarrow \pi_1(X/G,p(x_0))$?

Let $G$ be a discontinuous group (this means that it acts discontinuously with finite stabilizers) of homeomorphisms of a simply connected, locally compact metric space $X$. Let $p:X\rightarrow X/G$ ...

**3**

votes

**0**answers

76 views

### Classification of manifolds with ${\rm Ric}\geq 0$ wrt fundamental group

Note that $n$-manifolds $M$ with ${\rm Ric}\geq 0$ has a fundamental group of polynomial growth of degree $\leq n$ (proof : use Bishop volume theorem).
(Here a group $\Gamma$ is said to have ...

**6**

votes

**1**answer

273 views

### Etale fundamental of a parahoric group scheme

Let $p:X\rightarrow Y$ be a double cover of curves, denote by $$SU_n:=(p_*SL_n(\mathcal O_X))^{\tilde{\sigma}}$$
i.e. the $\tilde{\sigma}-$invariant part, the action of $\tilde{\sigma}$ is given by $$\...

**0**

votes

**0**answers

118 views

### A simple fundamental group of an hypersurface

Is there an example of analytic hypersurface in $C^n$ such that its fundamental group is simple i.e. does not have normal subgroups except the trivial group and the group itself ?
Thank you
EDIT : ...

**5**

votes

**1**answer

325 views

### What is the algebraic fundamental groups of $SO(n)$ and $Sp(2n)$?

Let $k$ be an algebraically closed field of characteristic zero. and let $$\sigma: SL_n(k)\rightarrow SL_n(k)$$
be an involution.
My questions are:
How could one calculate the fundamental group of $...

**26**

votes

**2**answers

1k views

### What is the fewest number of points you must delete from $\mathbb{R}^3$ to make it not simply connected?

This question concerns a set-theoretic aspect that I found interesting in the recent question asked by user Nick R., namely, Is
$\mathbb{R}^3\setminus\mathbb{Q}^3$ simply connected? He had asked ...

**39**

votes

**1**answer

1k views

### 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 ...

**7**

votes

**1**answer

262 views

### Does there exist an algebraic space with large fundamental group but no finite etale covers by schemes

Does there exits a smooth proper algebraic space $X$ over $\mathbb C$ with "large" fundamental group such that no finite etale cover of $X$ is a scheme?
By "large" fundamental group I mean that $X$ ...

**11**

votes

**2**answers

506 views

### Relationship between étale and topological $K(\pi,1)$s

I was trying to find a proof, or a counterexample to the claim that if $X/\mathbb{C}$ is connected smooth projective, then $X$ is a $K(\pi^{\mathrm{\acute{e}t}},1)$ if and only if $X^\mathrm{an}$ is a ...

**0**

votes

**1**answer

99 views

### 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 ...

**3**

votes

**1**answer

175 views

### Etale Fundamental group of an algebraic group

I want to calculate the algebraic fundamental group of a an algebraic group over a riemann surface over $\mathbb C$ (or a smooth algebraic projective curve). Let me state the first case where $\...

**2**

votes

**0**answers

152 views

### A homomorphism in the long exact sequence of a fibration for a homogeneous space of a Lie group

Let $G$ be a connected Lie group, and let $H\subset G$ be a (closed) Lie subgroup, not necessarily connected. Set $X=G/H$.
The fibration $j\colon G\to X$ with fiber $H$ induces an exact sequence
$$
\...

**2**

votes

**0**answers

131 views

### How do we see the rank of the braid group?

The only presentation of the braid group that most people ever see is the standard Artin presentation
$$B_n=\langle σ_1,\cdots,σ_{n−1}|\ σ_iσ_j=σ_jσ_i\ \ (|i−j|>1),\ σ_iσ_{i+1}σ_i=σ_{i+1}σ_i σ_{i+...

**17**

votes

**0**answers

307 views

### 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 ...

**0**

votes

**1**answer

87 views

### Double coset separability and the existence of vanishing sequences for surface group

Definition: Let $G$ be a group. $G$ is said to be double coset separable if given any finitely generated subgroups $H$ and $K$ in $G$, given any $g\in G$ and $h\not\in HgK$, there exists a finite ...

**5**

votes

**0**answers

168 views

### Does the fundamental group identify group structure on subvarieties of products of curves?

Let $C_1,\dots, C_n$ be smooth curves over $\overline{\mathbb F}_p$, not necessarily proper. Let $X$ be a subvariety of $C_1 \times \dots \times C_n$. I'm interested in the natural map:
$$ \pi_1^{ab}(...

**4**

votes

**1**answer

530 views

### A lower-dimensional algebraic topology problem between homology group and fundamental group

Let
\begin{equation}
A\stackrel{\alpha}{\longrightarrow}B\stackrel{\beta}{\longrightarrow}C\quad\quad (1)
\end{equation}
be a short sequence of (abelian or nonabelian) groups and homomorphisms. We say ...

**18**

votes

**1**answer

904 views

### 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 ...

**6**

votes

**1**answer

250 views

### Are profinite groups of cardinality $|\mathbb{R}|$ determined by their finite quotients?

Question: Let $G,H$ be profinite groups of cardinality $|\mathbb{R}|$, with the same finite quotients (here I only consider quotients by normal, open subgroups). Then are $G$ and $H$ isomorphic?
...

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votes

**0**answers

564 views

### Etale fundamental group of a curve in characteristic $p$

Let $C$ be a connected, smooth, proper curve of genus $g$ over an algebraically closed field $k$ of characteristic $p>0$. Let $\pi_1(C)$ be the etale fundamental group of $C$ - I only care about ...

**3**

votes

**1**answer

178 views

### Is the fundamental group of an open arithmetic Riemann surface contained in $\Gamma(2)$

Let $X$ be a non-compact Riemann surface with universal covering $\mathbb H$ and suppose that the fundamental group of $X$ is an arithmetic subgroup of $\mathrm{Aut}(\mathbb H) = \mathrm{PSL}_2(\...

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votes

**2**answers

1k views

### 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 ...

**1**

vote

**1**answer

239 views

### Algebraic topology, Dynamical systems [closed]

Let $T^2$ be a 2-torus and $f:T^2\rightarrow T^2$ a smooth map. Let $f_*:\pi_1(T^2)\rightarrow\pi_1(T^2)$ be the induced map on the fundamental group $\pi_1$. If $f_*$ has no eigenvalue greater than 1,...

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vote

**0**answers

168 views

### which sections of elliptic curves are conjugate?

Suppose you have a relative elliptic curves $f : E\rightarrow S$ (say $S$ is connected). Then suppose you have two sections $g,g' : S\rightarrow E$, corresponding to two sections $g_*,g'_*$ to the map ...

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votes

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153 views

### Algebraic fundamental group without regularity at infinity

Suppose $X$ is a smooth (connected) variety over $\mathbb{C}$. Let $\mathscr{C}$ be the category of finite rank vector bundles on $X$ equipped with an integrable connection, and let $\mathscr{C}'$ be ...

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vote

**0**answers

213 views

### Question about the specialization map for Etale Fundamental Groups

Let $A$ be a complete, discrete valuation ring, and let $s$ (resp $\eta$) be the special (resp. generic) point of $S=Spec(A)$. Let $\phi:X \rightarrow S$ be a proper morphism and fix geometric base ...

**1**

vote

**3**answers

314 views

### question about the induced homomorphism of etale fundamental groups

Background/Setup
For any connected scheme $S$, let $\text{FEt}_S$ denote the category of finite etale $S$-schemes. Let $f : X\rightarrow Y$ be a morphism of connected schemes, then for any finite ...

**8**

votes

**3**answers

2k views

### 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 $Top_*$.
...

**4**

votes

**1**answer

391 views

### Question about the fundamental group of rational homology 3-spheres

By a rational homology 3-sphere, I mean a compact oriented manifold three-manifold $Y$ with $H_1(Y)$ finite. My question is whether there exists a reasonable classification of such manifolds such that ...

**5**

votes

**1**answer

577 views

### On the fundamental group of closed 3-manifolds

I know that every finitely presented group can be realized as the fundamental group of a compact, connected, smooth manifold of dimension 4 (or higher). In dimension 2 there are strong restriction on ...

**4**

votes

**2**answers

341 views

### a question on rank of fundamental group

Assume $G$ be the fundamental group of a closed orientable hyperbolic 3-manifold. Let
$G_{1} = \langle a_{1},...,a_{k} \rangle$ be a free subgroup of $G$, and let $G_{2}=\langle a_{k+1} \rangle$ be a ...

**4**

votes

**1**answer

176 views

### Isomorphism étale fundamental group

Given a birational proper morphism $f\colon X \rightarrow Y$
( Assume $X$ and $Y$ irreducible )
of complex algebraic varieties.
It is always true that $f^* \colon \pi^{et}_1 (X)\rightarrow \pi^{et}_1(...

**4**

votes

**2**answers

300 views

### Fundamental group of a manifold with an $S^1$-action

Let $M$ be a compact connected manifold with an $S^1$-action. Suppose that $S^1$ has a fixed point in $M$. Is it true that $\pi_1(M)=\pi_1(M/S^1)$?
I is there some reference or a short proof of this ...

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votes

**0**answers

128 views

### Exact sequence of the fundamental group of the general fiber

Let $f\colon X\rightarrow Y$ be a morphism of complex algebraic varieties.
Let $y\in Y$ be a general point, then we have a sequence of homomorphisms
of fundamental groups induced by the inclusion of ...

**1**

vote

**0**answers

110 views

### algebraic varieties whose fundamental group is subgroup separable wrt subvariety subgroups

Call an algebraic variety $\pi_1$-subgroup separable iff,
for every $Y\subseteq X$ a closed subvariety and $\hat Y\xrightarrow{i} Y$ a normalisation of $Y$,
and subgroup $\Gamma=Im(\pi_1(\hat Y,y)\...

**4**

votes

**0**answers

112 views

### 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**

votes

**1**answer

211 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) \...

**1**

vote

**1**answer

353 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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531 views

### 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:...

**1**

vote

**1**answer

223 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 ...

**1**

vote

**1**answer

122 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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**0**answers

127 views

### 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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541 views

### 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 ...

**0**

votes

**1**answer

241 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 (...