# Questions tagged [fundamental-group]

The fundamental-group tag has no usage guidance.

233
questions

6
votes

3
answers

694
views

### 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. "...

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:
...

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 ...

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 ...

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. ...

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 ...

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, $$ (...

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 $\pi_1$-...

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 ...

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^{...

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,...

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\...

3
votes

0
answers

47
views

### 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 $L$ ...

7
votes

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 ...

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(\...

4
votes

0
answers

47
views

### 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+...

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 \...

0
votes

1
answer

78
views

### 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) ...

0
votes

0
answers

276
views

### 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 ...

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 ...

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 ...

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 ...

1
vote

0
answers

126
views

### 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 $\mathbf{P}^2_{\mathbf{F}_q}$?

2
votes

1
answer

160
views

### 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 ...

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 ...

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 $S^1 \times D^2$?

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\...

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 $\...

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 ...

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 $C_m(\tilde X)$ ...

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}, \...

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*}
...

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 $\overline{K}$-...

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 ...

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?

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{...

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 ...

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\...

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 ...

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 $G$-...

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 ...

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 ...

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 ...

1
vote

0
answers

64
views

### 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 ...

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 ...

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 ...

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 ...

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{...

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 ...

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 ...