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5
votes
0answers
144 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: $$ ...
2
votes
0answers
156 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 groups and homomorphisms. We say that the ...
6
votes
0answers
322 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
1answer
220 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? ...
13
votes
0answers
362 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
1answer
150 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) = ...
18
votes
2answers
819 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
1answer
174 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
vote
0answers
162 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 ...
5
votes
0answers
116 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 ...
1
vote
0answers
116 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 ...
0
votes
3answers
266 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 ...
4
votes
3answers
975 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
1answer
245 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 ...
4
votes
1answer
368 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
2answers
316 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
1answer
158 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 ...
4
votes
2answers
268 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 ...
3
votes
0answers
105 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
0answers
99 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 ...
4
votes
0answers
98 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
1answer
189 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
1answer
282 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 ...
20
votes
0answers
513 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 ...
1
vote
1answer
196 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
1answer
117 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 ...
4
votes
0answers
106 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?
11
votes
0answers
482 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
1answer
171 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 ...
2
votes
1answer
309 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 ...
3
votes
2answers
432 views

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 ...
7
votes
1answer
351 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
351 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 $\pi_1(X) \subset Aut(\tilde X)$. ...
6
votes
3answers
436 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 ...
5
votes
2answers
309 views

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) ...
2
votes
1answer
172 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
votes
1answer
193 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
votes
1answer
500 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 ...
12
votes
2answers
704 views

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 ...
8
votes
1answer
409 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 ...
1
vote
1answer
269 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 ...
1
vote
1answer
351 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 ...
2
votes
1answer
190 views

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 $\pi_1(\widetilde{X/G})$ ...
2
votes
2answers
303 views

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. ...
0
votes
1answer
432 views

fundamental group of a compact manifold

why fundamental group of of compact manifold is finitely presented
5
votes
1answer
241 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
394 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 ...
21
votes
2answers
791 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? ...
3
votes
1answer
299 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 ...
2
votes
1answer
355 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 ...