# Questions tagged [fundamental-group]

The fundamental-group tag has no usage guidance.

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### Approximation of homotopy avoiding a point in $\mathbb{R}^3$

For a proof that $\mathbb{R}^3\setminus \mathbb{Q}^3$ is simply connected using Baire category theorem I need to approximate an homotopy $H : [0,1]\times \mathbb{S}^1 \to \mathbb{R}^3$ from a loop $\...

**4**

votes

**1**answer

83 views

### Geodesic representatives in the orbifold fundamental group

Does every element in the orbifold fundamental group $\pi_1^{orb}(X,x)$ of a closed hyperbolic 2-orbifold $X$ admit a unique geodesic arc representing it?
Does every free homotopy class in $X$ admit ...

**4**

votes

**1**answer

134 views

### Invariant lifts of a closed curve on a surface of genus > 1

I am learning some things about surfaces of genus greater than $1$, and I am trying to answer this question :
Let $S$ be a compact and orientable surface of genus $g \geq 2$, and $c$ a closed curve ...

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**0**answers

248 views

### Relationships among constructions of fundamental group for schemes

There seem to be several constructions of fundamental group for schemes and stacks: by Grothendieck, Deligne, Nori, Noohi, Esnault-Hai, Vakil-Wickelgren, perhaps others as well. I am trying to ...

**5**

votes

**2**answers

307 views

### Galois categories for topological spaces?

Can the theory of Galois categories (as developed in SGA1) be modified to produce the usual fundamental group of a topological space (maybe assumed to be path connected and locally path connected)?
...

**15**

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**0**answers

443 views

### What would be the simplest analog of Langlands in algebraic topology?

It is oversimplified, I know, but just as a superficial analogy, one may think of the fact that abelianization of the fundamental group is the first homology group, as some remote relative of class ...

**3**

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**1**answer

100 views

### Multivariate Alexander polynomial vs single variable (Conway) Alexander polynomial

I consider the multivariate Alexander polynomial $\Delta(t_1,\ldots,t_n)$ for a $n$-component link (defined using e.g. the Fox derivative).
If we wish to construct a 1-variable polynomial $A(t)$, we ...

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593 views

### Can one compute the fundamental group of a complex variety? Other topological invariants? [duplicate]

Given a system of polynomial equations with rational coefficients, is there an algorithm to compute the geometric fundamental group of the variety defined by these equations? I'm interested in both ...

**3**

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**1**answer

388 views

### A projective (or free) $\mathbb{Z}\pi_1$-module

Suppose that $Z$ is a finite wedge of spheres containing circles and there exist maps $f:Y\to Z$ and $g:Z\to Y$ so that $g\circ f\simeq 1_Y$. Assume that there exists a map $h:X\to Y$ which induces ...

**6**

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**1**answer

113 views

### Example similar to the Griffiths twin cone but with fundamental group that allows surjection onto $\mathbb Z$

The Griffiths twin cone is an example of a wedge sum of two contractible spaces being non-contractible. Namely, it is the wedge sum $\mathbb G=C\mathbb H\vee_p C\mathbb H$ of two coni over the ...

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151 views

### Fundamental group of moduli space of K3's

According to Rizov (https://arxiv.org/abs/math/0506120), the moduli stack of primitively polarized K3 surfaces of degree 2d $\mathcal{M}_{d}$ is a Deligne-Mumford stack over $\mathbb{Z}$. I'm looking ...

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158 views

### Geometric fundamental group and algebraically closed residue field

my questions relates to the following talk of Tsuji:
https://www.youtube.com/watch?v=2brDj26phP0
At around 10:30 of the video, Tsuji is interrupted by a man stating that his construction does not ...

**6**

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**1**answer

179 views

### Finite étale covers of concentrated schemes and extension of base field

Let $k'/k$ be an extension of algebraically closed fields of characteristic $0$, and $X$ a concentrated (i.e. quasi-compact and quasi-separated) scheme over $k$.
Question: is the pullback functor ...

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### Is there a relationship between a quotient group of the fundamental group of X and the fundamental group of a quotient topology of X?

Let ($X$, $x_0$) be a topological space with a base point, and denote the fundamental group of $X$ as $\pi_1(X)$. Let $N$ be a normal subgroup of $\pi_1(X)$.
Does there necessarily exist an ...

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132 views

### Is the stack of smooth canonically polarized surfaces uniformisable (or a good orbifold)

This is a question about the fundamental group of the moduli of canonically polarized surfaces.
Let $\mathcal{M}$ be the (locally finite type separated Deligne-Mumford) algebraic stack of smooth ...

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votes

**1**answer

721 views

### Galois theory, topos vs fundamental groups

Classical Galois theory states that the etale topos X of a field k is equivalent to the classifying topos of the absolute Galois group of k.
(Marc Hoyois, Higher Galois theory, $\S$3, arXiv:1506....

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271 views

### Fundamental groups of non-orientable closed four-manifolds

The fundamental group of a closed orientable manifold is finitely presented, and every finitely presented group arises as the fundamental group of a closed orientable four-manifold; see this question. ...

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351 views

### Mapping class group and representation of fundamental group of Riemann surfaces

Let $S$ be a Riemann surface with genus $g>0$. Let $M$ be the mapping class group of $S$. $Hom(\pi_1(S),Gl(n, \mathbb{C}))$ is the representation space of fundamental group of $S$
Question: Is ...

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vote

**1**answer

199 views

### What does the group of automorphisms corresponding to $\mathfrak{g}$

I am reading a book titled "Lectures on An Introduction to Grothendieck's Theory of the Fundamental Group" by J.P. Murre. I am in the chapter 4 titled "Fundamental groups". Here he fixes a base ...

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**0**answers

130 views

### How does the fundamental group of $\mathbb G_{m,S}$ depend on the base scheme

Let $S$ be an integral noetherian regular scheme, and let $X =\mathbb{G}_{m,S}$.
How to compute $\pi_1^{et}(X)$?
Note. I am only interested in the part not coming trivially from the finite etale ...

**7**

votes

**1**answer

340 views

### Must an inverse limit of simply connected groups be simply connected?

While the fundamental group $\pi_1$ preserves products, it is not true in general that an inverse limit of simply connected topological spaces is simply connected. I would like to know if similar ...

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**1**answer

400 views

### Reconstruction of hyperbolic curves using the fundamental group

In the paper Curves and their Fundamental Groups written by Gerd Faltings, Mochizuki's proof of Grothendick's conjecture on anabelian curves is explained.
In the proof, he shows that for two ...

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**1**answer

478 views

### Simply connected slices

Assume $\Omega$ is an open set in $\mathbb R^3$
such that the intersection of $\Omega$ with any horizontal plane is simply connected.
Can you prove that $\Omega$ is simply connected?
(Note that ...

**2**

votes

**1**answer

245 views

### Does the fundamental group of the normalization of a scheme inject into the fundamental group of the scheme

Let $X$ be an integral noetherian finite type scheme over an algebraically closed field $k$. Let $X'\to X$ be its normalization. Is the induced homomorphism of etale fundamental groups $\pi_1(X')\to\...

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209 views

### On tangent space to the fundamental group scheme

Let $X$ be a smooth, projective complex curve of genus at least $2$. If I understand correctly, after choosing a base point, one can associate to $X$, a fundamental group scheme $\pi$. I am trying to ...

**7**

votes

**1**answer

205 views

### Do complex varieties have a dense open subset with residually finite fundamental group?

Let $S$ be a smooth connected variety over the complex numbers. The fundamental group might not be residually finite (i.e., the homomorphism $\pi_1(S(\mathbb C)) \to \pi_1^{\mathrm{et}}(S)$ might not ...

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260 views

### Monodromy representations are “quasi-unipotent”

Let $S$ be a smooth complex algebraic variety, let $b$ be a closed point of $X$, and let $f:X\to S$ be a polarized family of smooth projective varieties over $S$. This induces a monodromy ...

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votes

**1**answer

206 views

### Fundamental groups of open sets in $R^n$ with $n=3,4$

It is well known that every finitely presented group may be realised as fundamental group of some closed $4$-manifold.
What groups can be obtained as fundamental groups of open subsets of $R^4$? I'...

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386 views

### Is $H_{et}^1(X,F) = H^1(\pi_1^{et}(X), F(\bar{k}))$ true?

Let $X$ be a smooth geometrically connected scheme over a field $k$ of characteristic 0 (but not necessarily algebraically closed, I can take it to be a number field). Let $F$ be a finite algebraic ...

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443 views

### Finite covers of hyperbolic surfaces and the `second systole´

We are interested in the following ´relative´ version of residual finiteness for fundamental groups of surfaces. Similar discussions where given in this question: injectivity radius of hyperbolic ...

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votes

**1**answer

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### Images of boundary surfaces in 3-manifold groups

Let $M$ be a compact connected 3-manifold and let $S$ be a closed connected surface in $\partial M$. Let $G$ be the image of the map $\pi_1(S) \to \pi_1(M)$ induced by inclusion. I was reading the ...

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307 views

### Reference request: the comparison theorem for the étale fundamental group

I am looking for exact references for the comparison theorem for the étale fundamental group.
I mean the following result:
Theorem (Grothendieck). For a pointed algebraic variety $(X,x)$ over $\...

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406 views

### Are there any examples of hyperbolic curves over finite fields such that the action of frobenius on its prime-to-$p$ fundamental group is known?

Let $X$ smooth curve over a finite field $\mathbb{F}_q$ of type $(g,n)$ - that is, $X$ is an open subscheme of its genus $g$ compactification obtained by removing $n$ points.
Any such curve ...

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482 views

### Nodal curve in a smooth variety with injective map on fundamental groups

Let $C$ be the nodal curve obtained by gluing together the points $0$ and $1$ in $\mathbb{A}^1_{\mathbb{C}}$. The topological fundamental group of $C$ is isomorphic to $\mathbb{Z}$.
One can find an ...

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votes

**1**answer

209 views

### Fundamental groups of hyperbolic $4$-manifolds and $\rm CAT(0)$ cube complexes

Suppose $M^4$ is a compact hyperbolic (i.e. curvature $-1$) $4$-manifold and $\Gamma\cong\pi_1(M^4)$.
Is there any expectation whether $\Gamma$ acts properly and co-compactly on a $\rm CAT(0)$ cube ...

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272 views

### Is there a presentation to the kernel of the prime-to-$p$ fundamental short exact sequence of curves over finite fields?

Let $X$ be $\mathbb{P}^1_{\mathbb{F}_q}\smallsetminus \{a_1,...,a_r\}$, where $a_1,...,a_r$ are some $\mathbb{F}_q$-rational points. Let $\bar X :=X_{\bar{\mathbb{F}}_q}$. There is a short exact ...

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226 views

### Fundamental groups of complements of divisors in $\mathbb P^2$

Let $D$ be a divisor in $\mathbb P^2_{\mathbb C}$ and let $X= \mathbb P^2_{\mathbb C} - D$.
Under what condition on $D$ is the fundamental group of $X$ infinite?

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votes

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153 views

### Conjugacy-invariance of sections of etale homotopy exact sequence

My questions arise on page xiv of Stix's Rational Points and Arithmetic of Fundamental Groups. Here is an excerpt:
Given a geometrically connected variety $X/k$, a fixed separable closure $\bar{k}/k$,...

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votes

**1**answer

793 views

### What is the arithmetic fundamental group?

Let $X$ be an irreducible variety defined over $\mathbf{F}_p$. What's the "arithmetic fundamental group" of $X$? How does this relate to the algebraic fundamental group of a scheme? What's a good ...

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259 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)\...

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**1**answer

680 views

### Inverse galois problem and étale homotopy

Is there any relation between étale homotopy theory (Grothendieck-Galois theory) and the inverse Galois problem?...I mean...in classical homotopy theory, every finite group $G$ realizes as a "Galois ...

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votes

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715 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 ...

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111 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 ...

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### 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 ...

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906 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 ...

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260 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 $p:X\rightarrow X/G$ ...

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### 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 ...

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votes

**1**answer

298 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 $$\...

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**0**answers

138 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

475 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 $...