Questions tagged [fundamental-group]

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Motivation of the fundamental theorem of covering spaces

The fundamental theorem of covering spaces states that for a nice topological space $X$, there is an equivalence of categories between covering spaces over $X$ and left $\pi_1(X)$-sets. "...
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0 votes
0 answers
108 views

Can someone explain this proof on aspherical manifolds?

I am trying to understand this proof that the fundamental group of an aspherical manifold is torsion free. The proof is lemma 4.1 from Aspherical manifolds at the Manifold Atlas Project. The proof is: ...
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12 votes
2 answers
704 views

Topos-theoretic Galois theory

This page is an overview of some of the types of "Galois theories" there are. One of the most basic type is the fundamental theorem of covering spaces, which says, roughly, that for each ...
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5 votes
0 answers
79 views

Fundamental groups and cellular walks

Suppose $M$ is a smooth manifold (compact if desired) with a cell structure or other nice stratification. Call a path $\gamma : [0,1] \to M$ transverse to the stratification if there is a finite ...
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3 votes
0 answers
122 views

Relationship between the holonomy pseudogroup and holonomy homomorphism (foliation)

This question is surely a duplication of https://math.stackexchange.com/questions/4343635/relationship-between-the-holonomy-pseudogroup-and-holonomy-homomorphism-foliati , however, I got no replies. ...
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6 votes
2 answers
280 views

Do surface groups embed into PSL_2 over a real quadratic integer ring?

$\DeclareMathOperator\PSL{PSL}$ Let $ \mathbb{Z} $ be the ring of integers and $ \mathbb{R} $ the field of real numbers. Let $ \Sigma_g $ be a surface of genus $ g \geq 2 $. Let $ \pi_1(\Sigma_g) $ be ...
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0 votes
0 answers
51 views

How to show that, $ C_c ( \pi_{1}^{ \mathrm{top} } (X) ) $ is dense in $ C_0 (X/G) $?

Let $G$ be a locally compact, Hausdorff, second countable, topological group. Let $X$ be a locally compact, Hausdorff $G$-space. The action of $G$ on $X$ gives an action of $G$ on $ C_0 (X) $ by, $$ (...
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7 votes
2 answers
875 views

Does the isomorphic of the fundamental groups imply the existence of a mapping inducing an isomorphism?

A pair of continuous mappings $f \colon X \to Y$ and $g \colon Y \to X$ is called $\pi_1$-equivalence if they induce mutually inverse isomorphisms of fundamental groups. Spaces are called $\pi_1$-...
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4 votes
2 answers
372 views

Can the loops in the definition of the fundamental group be considered injective?

Let $\mathrm{С}$ be some class of topological spaces that includes at least all subspaces of $\mathbb{R}^n $. Further we are in the category $\mathrm{С}_{*}$ (the category of point spaces; all ...
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3 votes
0 answers
66 views

Fundamental group of hyperbolic 2-orbifold

Suppose $\Gamma$ is a cocompact lattice of $PSL_2(\mathbb{R})$. Then $\mathbb{H}^2/\Gamma$ has a natural structure of orbifold. My questions are: What is $\pi_1(\mathbb{H}^2/\Gamma)$? What is $\pi_1^{...
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7 votes
2 answers
317 views

Fundamental group of the space of smooth embeddings of $S^1$ into $\mathbb R^3$

Has the fundamental group of the space of smooth embeddings of $S^1$ into $\mathbb R^3$ been completely computed? Say the basepoint is an unknot. Maybe something is known for other components? If yes,...
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6 votes
0 answers
188 views

Classifying nested 3-manifolds with fundamental group property

Let $M_1\subseteq M_2\subseteq\mathbb R^3$ be closed connected subsets with smooth boundary. Suppose that every closed loop in $M_1$ is freely homotopic inside $M_2$ to a closed loop inside $M_2\...
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3 votes
0 answers
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What's the Milnor's link group for the trivial knot in a lens space?

For a link $L$ in a 3-manifold $Y$, Milnor's paper "Link Groups" https://link.springer.com/content/pdf/10.1007/BF01393902.pdf defined the link group as some quotient of $\pi_1(Y-L)$. If $L$ ...
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  • 385
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1 answer
551 views

Representation theory of higher homotopy groups

I've seen some works on the representation of fundamental groups, which are (at least for me) quite important topic in mathematics. For example, Riemann-Hilbert correspondence relates representation ...
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4 votes
1 answer
133 views

What is the minimal length of a “Diagonal” in a Torus?

Given a Riemannian torus $(T,d)$ with fundamental group $\pi_1(T)=\langle a,b \mid ab=ba \rangle$. Denote for any $\gamma \in \pi_1(T)$ the infimum length of all representatives of $\gamma$ by $L(\...
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4 votes
0 answers
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Fundamental group of the complement of some quadric cones

cross-posting from MathSE Problem Consider the domain $$\Omega=\mathbb{C}^4\setminus\{z_0(z_1^2+z_2^2+z_3^2)=0\}$$ and the map $$F:\Omega\to\mathbb{CP}^1\qquad F(z_0,z_1,z_2,z_3)=[z_0^2:z_1^2+z_2^2+...
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  • 1,165
4 votes
1 answer
293 views

Surface bundles associated to a short exact sequence of groups

Suppose $S$ is a closed, connected, oriented surface of genus at least two and $G$ is any group. Suppose further that $\Gamma$ is any group that fits into the following short exact sequence: $$ 1 \to \...
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  • 293
0 votes
1 answer
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Fundamental group to groupoid : bijection between homotopy classes?

I'm looking at the fundamental group $\pi_{1}(M)$ of the $n^{th}$ unordered configuration space $M$ of $\mathbb{R}^{d}$. In particular, it's well-known that $\pi_{1}(M)\cong S_{n}$ (symmetric group) ...
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Is $\operatorname{Aut}(\mathcal{M})$ a fundamental group in Grothendieck's sense?

This question is a follow-up to Are there infinitely many L-rigs? and to Is an automorphic form of $\operatorname{GL}_{n}(\mathbb{A}_{\mathbb{Q}})$ determined by its L-function?. I copy paste a deepl ...
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43 votes
3 answers
4k views

"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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3 votes
1 answer
221 views

Fundamental group of twisted loop space

I'm interested in computing the fundamental group of the twisted loop space $$\Omega_f(M)=\{ \gamma \in C^{\infty}(\Bbb R,M) \mid \gamma(s+1)=f\gamma(s)\}$$ where $f \in \text{Aut}(M,x_0)$, for ...
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  • 503
2 votes
0 answers
160 views

Isocrystals on simply connected varieties

Esnault and Shiho - Convergent isocrystals on simply connected varieties proves that there are no non-trivial convergent isocrystals on simply connected varieties. There is another similar result in ...
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1 vote
0 answers
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What is the étale fundamental group of projective spaces over finite fields?

Is there any convenient way to understand the étale fundamental group of projective spaces over finite fields, in particular, the étale fundamental group of $\mathbf{P}^2_{\mathbf{F}_q}$?
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2 votes
1 answer
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Lifting of a proper map in the cover is a proper map

Let $M$ be an orientable surface without boundary$($I am not assuming $M$ is compact, it can be non-compact$)$. Let $\Phi: M\to M$ be a proper homotopy-equivalnce$($A proper homotopy-equivalence can ...
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  • 265
2 votes
1 answer
190 views

Does the Hawaiian Earring Group embed into the permutation group of $\mathbb N$?

Recall that the Hawaiian earring group, $\mathbb G$, is the fundamental group of the Hawaiian Earing using the point at the origin. It can be understood more combinatorially as a subgroup of the ...
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5 votes
2 answers
675 views

3-manifold with fundamental group $\mathbb Z$

Let $M$ be a compact $3$-manifold with nonempty boundary. If $\pi_1(M)=\mathbb Z$, can we prove that $M$ is homeomorphic to $S^1 \times D^2$?
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  • 2,465
3 votes
0 answers
125 views

Is the category of covering spaces always a topos?

It is well knows that for a nice (locally path connected, semi-locally simply connected) topological spaces, the category of covering spaces over $X$ is equivalent to the functor category $\left[\Pi_1\...
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6 votes
0 answers
220 views

Fundamental group of a product in characteristic 0

It is proven in SGA1 that if $k$ is an algebraically closed field, if $X$ is a proper $k$-scheme and if $Y$ is a locally noetherian $k$-scheme (say, $X$ and $Y$ are non-empty and connected) then $\...
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5 votes
1 answer
231 views

triviality of homology with local coefficients

Let $X$ be a manifold or a CW-complex. Let $\pi: \tilde X\longrightarrow X$ be a covering map. Let $\pi_1(X)$ be the fundamental group of $X$ and let $\rho: \pi_1(X)\longrightarrow O(n)$ be an ...
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2 votes
1 answer
202 views

can the actions of fundamental groups annihilate homology?

Let $X$ be a path-connected manifold (or a CW complex). Let $\pi_1(X)$ be the fundamental group of $X$. Let $\pi: \tilde X\longrightarrow X$ be a covering map. For each $m\geq 0$, let $C_m(\tilde X)$ ...
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  • 1,940
6 votes
1 answer
141 views

Representation of fundamental groupoid as $2$-sheaf

By https://arxiv.org/abs/1406.4419 (The fundamental groupoid as a terminal costack, Ilia Pirashvili), we know that for a topological space $X$, the $2$-functor $$\text{Top}(X)\rightarrow \text{Gpd}, \...
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2 votes
0 answers
130 views

Outer Galois representations and Tate modules of Jacobian varieties

Let $X$ be a proper smooth curve over a field $k$. Then we have an exact sequence of profinite groups \begin{equation*} 1 \to \pi_1(X_{\overline k}) \to \pi_1(X) \to G_k \to 1, \end{equation*} ...
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9 votes
3 answers
995 views

Are "large enough" finite etale covers arithmetic?

Let $X$ be a variety over a number field $K$. Then it is known that for any topological covering $X' \to X(\mathbb{C})$, the topological space $X'$ can be given the structure of a $\overline{K}$-...
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5 votes
0 answers
364 views

Complex conjugation inducing a trivial map on the fundamental group

Let $V$ be a smooth projective complex variety defined over the reals such that $G=\pi_1(V)$ is a non-abelian finite simple group. Assume that $V$ has a real point. Can the map $G\to G$ induced by ...
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3 votes
1 answer
279 views

A complex variety with a finite non-abelian simple fundamental group

Does there exist a complex smooth proper variety whose fundamental group is finite non-abelian simple?
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1 vote
0 answers
41 views

Restricted wreath product as fundamental group of a space with coinciding Reidemeister and Nielsen numbers

I am studying a group $\mathbb{Z}_n \wr \mathbb{Z}^k$, where $\wr$ denotes the restricted wreath product: $$ \mathbb{Z}_n \wr \mathbb{Z}^k = \bigoplus_{x\in\mathbb{Z}^k}(\mathbb{Z_n})_x\rtimes\mathbb{...
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3 votes
0 answers
160 views

Galois theory of ramified coverings vs classical Galois theory

That's an exact copy of my former MSE question I asked a couple of weeks ago and unfortunately not got the answer I was looking for. The question adresses reuns' answer in this thread: Algebraic ...
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6 votes
2 answers
842 views

Action of fundamental group on homotopy fiber

For a Serre fibration of pointed topological spaces $f:X \to B$, there is an action of $\pi_1\left(B,b_0\right)$ on the fiber $F$. The construction of this action I'm familiar with uses a lift $F\...
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6 votes
1 answer
288 views

An extension of symplectomorphism group

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Sp{Sp}$Let $\omega=\sum dx_i\wedge dy_i$ be the standard symplectic structure of $\mathbb{R}^{2n}=\mathbb{R}^{n}\times \mathbb{R}^n$. We consider the ...
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2 votes
1 answer
171 views

Čech cocycles and monodromy

It is well known that over a topological space $X$ (and choosing an open cover $\mathfrak{U}$) every locally constant Cech cocycle $g$ on $\mathfrak{U}$ with coefficients in a group $G$ yields a $G$-...
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7 votes
1 answer
335 views

Understanding fundamental group of Poincare homology sphere

I'm currently reading Knots, Links, Braids, and 3-Manifolds by V. V. Prasolov and A. B. Sossinsky. I have trouble understanding the following picture. The dashed line denotes a trefoil whose tubular ...
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4 votes
0 answers
262 views

Contractibility and orientation double cover

Question. Let $M$ be a triangulated non-orientable 3-manifold with non-orientable boundary. (It is possible to assume that the boundary is the Klein bottle.) Let $\ell$ be a non-orientable loop on the ...
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1 vote
2 answers
290 views

Non-self-intersecting paths on $\mathbb{C}\setminus\{0,1\}$ [closed]

Let us make two small holes around points $0$ and $1$ on the complex plane and consider non-self-intersecting paths that start on the boundary of one hole and finish at the boundary of the another. It ...
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  • 29
1 vote
0 answers
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Glueing local systems over union of compact Riemann surfaces

Let $X,Y$ be two connected, non-singular compact Riemann surfaces such that $X$ intersects $Y$ transversely at two distinct points. Let $L$ be a $\mathbb{C}$-local system on $X$. Let $L'$ be the ...
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  • 1,533
1 vote
0 answers
128 views

Canonical étale path between a point and its ''nearby'' point

Consider the punctored line $X=\Bbb{A}^1_k\setminus \{s_1,\ldots,s_n\}$ over some field $k$. A(n étale) path in $X$ between two geometric points $x$ and $y$ is, by definition, an isomorphism between ...
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  • 271
8 votes
1 answer
322 views

The direct product of the geometric fundamental group and the absolute Galois group

Given a geometrically connected variety $X$ over $\mathbb{Q}$ we have a short exact sequence $$ 1\to \pi_1(X_{\overline{\mathbb{Q}}})\to \pi_1(X)\to Gal(\overline{\mathbb{Q}}/\mathbb{Q})\to 1. $$ A ...
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18 votes
2 answers
766 views

Fundamental group of punctured simply connected subset of $\mathbb{R}^2$

(This question is originally from Math.SE where it was suggested that I ask the question here) Let $S$ be a simply connected subset of $\mathbb{R}^2$ and let $x$ be an interior point of $S$, meaning ...
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5 votes
2 answers
448 views

Can we define fundamental groups functorially for non-pointed path connected topological spaces?

Let $\text{ppTop}$ denote the category of pointed and path connected topological spaces with morphisms base-preserve continuous maps. The fundamental group gives a functor $FG: \text{ppTop}\to \text{...
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  • 8,079
11 votes
2 answers
232 views

Fundamental group under Gelfand duality

Gelfand duality states that the functor of continuous functions $C(-)$ from compact Hausdorff topological to commutative $C^*$-algebras is an equivalence of categories. In other words, all topological ...
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1 vote
1 answer
204 views

Fundamental group of the complement of cell subcomplexes

Given a regular CW complex stucture on a manifold $C$ of dimension $n$ and a subcomplex $D$ of dimension $n-2$, I want to compute the fundamental group of the complement $\pi_1(C\setminus D)$. A ...
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