# Questions tagged [fundamental-group]

The tag has no usage guidance.

233 questions
Filter by
Sorted by
Tagged with
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. "...
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: ...
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 ...
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 ...
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. ...
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 ...
51 views

293 views

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*} ...
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 $\overline{K}$-...
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 ... 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
41 views

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 ... ### 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 ...