# Questions tagged [fundamental-group]

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### Reliable literature with the list of centers of all simply connected simple real Lie groups

Wikipedia webpage https://en.wikipedia.org/wiki/Simple_Lie_group contains a full list of all simple (centerless) real Lie groups. One of the columns in tables (therein) contains fundamental groups of ...
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### Fundamental group of a quotient by a group action

Suppose I have a quotient $X \to S$ by a finite abelian group $G$ action (I have several cases, but in all of them the group $G$ and the action could be written explicitly), where $X,S$ are surfaces (...
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### Example of three dimensional atoroidal Poincaré duality group with some pathology

I am looking for a 3-manifold which is closed, aspherical, orientable, and atoroidal. And additionally I want to see an example that does not admit a fixed-point-free action on a simplicial tree. As a ...
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### A question on the manifold $\{n\otimes n-m\otimes m:n,m\in S^2,(n,m)=0\}$

Consider a manifold $N$ defined as follows $$N=\{n\otimes n-m\otimes m:n,m\in S^2,\quad(n,m)=0\}\subset M^{3\times 3},$$ where $S^2$ denotes the two dimensional sphere, $(\cdot,\cdot)$ ...
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### Extending étale covers from the regular locus to a resolution of singularities

Let $X$ be a normal proper variety with rational singularities (or terminal if that is necessary) and $X_{\text{reg}} \to X$ the regular locus. Let $\pi : \tilde{X} \to X$ be a resolution of ...
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### For which spaces $S^n$ ($n\geq 2$) is a universal covering space?

I know that $S^n$ $(n\geq 2)$ is a universal covering space for itself and $\mathbb{RP}^n$. But my question is, for which spaces (up to homotopy equivalence) is $S^n$ ($n\geq 2$) a universal covering ...
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### Does this sequence stop?

Let $\{ X_i\}$ ($i=1,2,\ldots$) be a family finite CW-complexes such that $X_{i+1}$ is homotopy domintaed by $X_i$, i.e. there exists contionuos maps $g_i:X_i \to X_{i+1}$ and $f_i :X_{i+1} \to X_i$ ...
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### Behaviour of cycles modulo algebraic equivalence on an etale covering

I found a neat result in Beauville's paper "VARIÉTÉS DE PRYM ET JACOBIENNES INTERMÉDIAIRES" : if $U \subset \mathbb{P}^n$ is an open and $V \to U$ is a conic bundle whose fibres are all ...
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### Motivation of Zariski–Van Kampen theorem

The Zariski–Van Kampen theorem gives the presentation of the fundamental group of the complement of the plane curve of degree $d$. But what's the motivation of this theorem? More generally, why are ...
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### Descent obstruction of an open curve in an elliptic curve

Let $E$ be an elliptic curve over a number field $k$, and for an extension $K/k$ we denote by $E_K$ the base change $E \times_k K$. By fixing an embedding $k \hookrightarrow \mathbb{C}$, the etale ...
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### Ways to prove that $n$-component Brunnian link is nontrivial

The attached image shows a way to construct an $n$-component Brunnian link for any $n\geq 3$. That is, this link is not trivial, but deleting any of its components makes the new link trivial. The ...
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Let $\Sigma \subset \mathbb{R}^3$ be a compact embedded surface with boundary $\partial \Sigma$ and $i:\Sigma\setminus \partial\Sigma \to \mathbb{R}^3 \setminus \partial\Sigma$ the inclusion. Is the ...
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### Unramified section associated to a rational point

This is a question for those familiar with the section conjecture, so I'll do away with the definition of a ramification map in this case. Here is the definition of a ramification map from an etale ...
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### Künneth formula for $\pi_1$-proper morphisms

Context: Let $X$ and $Y$ be connected qcqs schemes over an algebraically closed field $k$. Denote by $\pi_1(X)$, $\pi_1(Y)$ their étale fundamental groups (base points omitted). Grothendieck proved ...
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### The fundamental group of quotient space of 3-folds

Let $S$ be a K3 surface with an involution $\iota_S$, $E$ an elliptic curve with an involution $\iota_E$. Assume the fixed locus of $S$ under $\iota_S$ contains $N>0$ disjoint curves. Note the ...
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### Boundary of a $4$-manifold and the fundamental group

I am trying to learn $4$-manifolds with boundaries and I don't know much about this topic so these questions may be silly. Given a $4$-manifold $M$ with a boundary say $N$, Assume $\pi_1(N)$ is known,...
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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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