Questions tagged [fundamental-group]
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268 questions
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$P^1$ minus k points
For $k\geq 3$, and $k$ arbitrary points $S=( z_1,\cdots,z_k ) \in \mathbb{P}^1$, we can write
$$ P^1 \setminus S \cong \mathbb{H}/G $$
where $\mathbb{H}$ is the upper-half plane and $G\subset PSL(2,...
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When is the class of functions between sets a set?
I'm reading the paper 'The big fundamental group, big Hawaiian earrings and the big free groups'. The authors state that the class of homotopy equivalences of loops in the space he dubs as the big ...
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Is $\pi_1(\widetilde{X/G})$ always finite if $\pi_1(X)$ is finite?
Let $X$ be a smooth complex manifold with finite fundamental group. Suppose that a finite group $G$ acts on $X$ and let $\widetilde{X/G}$ be a resolution of singularities. Is $\pi_1(\widetilde{X/G})$ ...
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About "de-Rham" and "l-adic" local systems - comparison
Hello,
Suppose that $k$ is an algebraically closed field of char. 0.
Let $X$ be a smooth connected variety over $k$.
Then I have the category $A$ of Regular Singular smooth $D$-modules on $X$ (i.e. ...
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Fundamental group of a compact manifold
Why is the fundamental group of a compact manifold finitely presented?
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Conjugation of homogeneous spaces
Let $X$ be a smooth irreducible algebraic variety
over the field of complex numbers ${\mathbb{C}}$.
Let $x\in X({\mathbb{C}})$.
Let $\tau$ be an automorphism of ${\mathbb{C}}$ (not necessarily ...
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Fundamental Group and Etale Cohomology
I encountered the following statement without a reference many times. For a smooth variety $X$ over a perfect field $k$.
$Hom(H^1_{et}(X, \mathbb{Z}/n), \mathbb{Z}/n) \cong \pi^{ab}_1(X)/n$
Is there ...
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Profinite groups as étale fundamental groups
Does every profinite group arise as the étale fundamental group of a connected scheme?
Equivalently, does every Galois category arise as the category of finite étale covers of a connected scheme?
...
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Abelianized fundamental group of a curve over a finite field
Let $X$ be a smooth, projective, and geometrically connected curve over a finite field $\mathbb{F}_q$ and fix a geometric point $\overline{x} : \text{Spec } \overline{\mathbb{F}_q} \to X$. Then there ...
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Grothendieck's section conjecture and base change: restricting sections
Let $X$ be a smooth projective geometrically connected curve over $\mathbf{Q}$ of genus at least two. Fix an algebraic closure $\overline{\mathbf{Q}}$ of $\mathbf{Q}$ and let $G_{\mathbf{Q}}$ be the ...
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local fundamental group of elliptic singularities
Is the local fundamental group of an elliptic singularity virtually solvable ? Here (the terminology is sometimes divergent) an elliptic singularity is a (germ of) normal surface $(X,x)$ such that $X$ ...
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Is the fundamental group functor a left-adjoint?
Theorem 1B.9 in Hatcher's Algebraic Topology says that for a (pointed) connected CW complex $X$ and group $G$, there is a bijection $\text{Hom}(\pi_1(X), G) \cong [X,K(G,1)]$, where $\pi_1(X)$ is the ...
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Is every ''group-completion'' map an acyclic map?
I start with a longer discussion which will result in a precise version of the question. I am puzzled about an issue with the
Quillen plus construction. I have seen outstanding experts being confused ...
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Manifolds with prescribed fundamental group and finitely many trivial homotopy groups
Fix $G$, a finitely generated presented group.
It is known that for every $k > 3$ there is a closed $k$-manifold whose fundamental group is $G$. Similarly, there is a topological space with ...
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How should one understand orbifold fundamental groups?
I am studying orbifold fundamental group (or more generally orbifold homotopy groups). In a nutshell, my questions is: what are they intuitively? In what follows I give definitions and more precise ...
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Discretifications of the fundamental group functor
Grothendieck calls a "discretification" of a profinite group $\widehat G$, a
discrete group $G$ whose profinite completion is isomorphic to $\widehat G$.
Does Grothendieck also define a notion of ...
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$\pi_1$ Sequence of Topological Groups
Consider a connected topological group $G$ (not necessarily Lie). You have some maps $G\times G\to G$, such as projection to either summand, or multiplication $(g,h)\mapsto gh$. Now let's look at a ...
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When is a three-manifold deck transformation group solvable?
Suppose that $\pi:Y \to Y'$ is a regular covering of closed, connected, orientable three-manifolds and let $G$ be the deck transformation group. Furthermore, suppose that $Y$ is a rational homology ...
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Codimension Two Embeddings in Goodwillie-Weiss Manifold Calculus, and the Difficulty of Fundamental Groups
In manifold calculus, there are various analyticity estimates which run into trouble for codimension two embeddings. For instance, the functor $\operatorname{Emb}(M,N)$ is analytic in $M$ if $\dim M \...
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connected compact semisimple lie group finite fundamental group
I was told that the fundamental group of a connected, compact, semisimple Lie group is finite, with the outline of a possible way to prove this fact. Is there any source however that fleshes this out ...
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A question about the Tannakian etale fundamental group of a curve
Let $X$ be a smooth connected quasi-projective curve over $\mathbf{Q}$. Let $U$ be the pro-unipotent etale fundamental group of $X$ over $\mathbf{Q}_p$.
$U^1 = U$ and let $U^n =[U,U^{n-1}]$.
Let $n\...
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Homology of Covering Spaces
Let $A$ be a subgroup of a group $G$. Then since $A$ is a subgroup of the fundamental group $\pi_1(K(G,1))=G$, there is a covering space $p\colon Y\to K(G,1)$ with $p_*(\pi_1(Y))=A$. So the homology ...
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fundamental groups of smooth projective variety.
Is there a discrete group G which is the fundamental group of a compact Kahler
manifold but which is not the fundamental group of any smooth projective complex algebraic variety?
It is known that ...
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What properties make $[0,1]$ a good candidate for defining fundamental groups?
The title essentially says it all. Consider the category $\mathfrak{Top}_2$ of triples $(J,e_0,e_1)$ where $J$ is a topological space, and $e_i \in J$. There is an obvious generalization of the ...
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Can we define homotopy groups using Tannakian categories
This is another vague question. Hope you guys don't mind.
Let $T$ be a Tannakian category. For any fibre functor $F$ on $T$ we define the fundamental group of $T$ at $F$, denoted by $\pi_1(T,F)$, to ...
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Number of connected components of the Hurwitz space $H_n^o$ and subgroups of the fundamental group
A cover (of $\mathbf{P}^1_{\overline{\mathbf{Q}}}$) is a finite morphism $X\to \mathbf{P}^1_{\overline{\mathbf{Q}}}$, where $X$ is a smooth projective connected curve over $\overline{\mathbf{Q}}$. ...
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Are acyclic subcomplexes of finite contractible 2-complexes contractible?
Let $Y$ be a contractible finite simplicial 2-complex.
Let $X$ be an acyclic subcomplex of $Y$ (i.e. $X$ connected, $H_1(X)=0$, $H_2(X)=0$).
Is $X$ contractible? (Equivalently, is $\pi_1(X)$ trivial?)...
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Geometric interpretation of the lower central series for the fundamental group?
For any group $G$ we can form the lower central series of normal subgroups by taking $G_0 = G$, $G_1 = [G,G]$, $G_{i+1} = [G,G_i]$. We can check this gives a normal chain
$$G_0 \ge G_1 \ge ... \ge G_i ...
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3-manifolds with solvable fundamental group
Is there a nice reference for the classification of closed 3-manifolds with solvable (nilpotent, abelian, etc.) fundamental group, assuming the Geometrization Conjecture?
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Tutte polynomials of appropriate Cayley graphs
I was quite intrigued by Tutte polynomials in a recent talk I had been to. It was introduced as a polynomial associated to a undirected finite graph. For a graph $G=(V,E)$ we form the polynomial
$T_G(...
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Analytic Characterization of Parallel Transport of Fundamental Groups
(Note that I've edited the main body of the question to make it clear for other readers.)
Fix a principal $G$-bundle $\rho: P \rightarrow X$ and fix a point $p \in P_x$ in the fiber above $x \in X$. ...
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What are the different theories that the motivic fundamental group attempts to unify?
I must preface by confessing complete ignorance in the subject. I've read introductory texts about the theory of motives, but I am certainly no expert.
In http://www.math.ias.edu/files/deligne/...
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Branch cuts of $GL_n^+(\mathbb{R})$
Branch cuts
Let $GL_n^+(\mathbb{R})$ denote the group of $n\times n$ real matrices with positive determinant. Topologically, $GL_n^+(\mathbb{R})$ is connected, and
$$ \pi_1(GL_2^+(\mathbb{R})) = \...
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Motives from the fundamental group made nilpotent
I am reading the fascinating paper of Deligne on "le groupe fondamental de la droite projective moins trois points", and other stuffs related to anabelian geometry. This suggested the following ...
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locally constant constructible sheaves and finite etale coverings
Maybe it is well known to experts or maybe it is just a stupid idea, but I will ask any way.
We know that if $X$ is a topological space, then there is an equivalence of categories between the ...
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Fundamental groups of surfaces
What are some properties that hold for the fundamental group of a surface and do not necessarily hold for the fundamental groups of manifolds of higher dimensions?
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Fundamental group of R^2-Q^2
After learning about the fundamental group, and proving that $\mathbb{R}^n$ minus any countable set is path-connected, I started wondering if the fundamental group of $\mathbb{R}^2-\mathbb{Q}^2$ is ...
- 3,079
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finite quotients of fundamental groups in positive characteristic
For affine smooth curves over $k=\bar{k}$ of char. $p,$ Abhyankar's conjecture (proved by Raynaud and Harbater) tells us exactly which finite groups can be realized as quotients of their fundamental ...
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generating the etale fundamental group by sections?
Let $X$ be a proper smooth scheme over a field $k$ of characteristic zero (well you can naturally weaken the assumption to normal integral scheme over some "nice" base like $\mathbb{Z}$, $\mathbb{F}_q$...
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Fundamental groups of closed hyperbolic 3-manifolds are freely indecomposable
I believe the following statement is true, and I've even seen it referenced here. Could someone point me to a proof?
The fundamental group of a closed hyperbolic 3-manifold is not a free product.
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Reference needed: Isomorphism on pi_1 and homology gives weak equivalence
Let $f : X \to Y$ be a map between a connected space $X$ and a space $Y$. If $\pi(f) : \pi_1(X) \to \pi_1(Y)$ is an isomorphism, and $H_n(f) : H_n(X, G) \to H_n(Y, G)$ is an isomorphism for all $n \ge ...
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Topological vs pro fundamental groups
Consider the following two structure-adding refinements of the fundamental group of a topological space:
the set $\pi_1(X)$ inherits a quotient topology from the compact-open topology of $X^{S^1}$, ...
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explicit linear representations of fundamental groups of surfaces
I am looking for an explicit representation of the fundamental group of a closed orientable surface of genus >1. I guess they should be abundant in degree 2. Did anyone see the explicit matrix ...
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What is the wild fundamental group?
In the abstract of
Singularités irrégulières Correspondance et documents
Pierre Deligne, Bernard Malgrange, Jean-Pierre Ramis
Documents mathématiques 5 (2007), xii+188 pages (link)
there is a ...
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What are Galois Categories used for?
Galois categories are introduced (for the first time?) in SGA1, but here's an English introduction that's available online: http://www.math.uchicago.edu/~may/VIGRE/VIGRE2009/REUPapers/Lynn.pdf
It ...
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Finite vector bundles over punctured affine spaces
Let $X$ be a connected scheme. Recall that a vector bundle $V$ on $X$ is called finite if there are two different polynomials $f,g \in \mathbb N[T]$ such that $f(V) = g(V)$ inside the semiring of ...
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Does the etale fundamental group of the projective line minus a finite number of points over a finite field depend on the points?
Clearly the etale fundamental group of $\mathbb{P}^1_{\mathbb{C}} \setminus \{a_1,...,a_r\}$ doesn't depend on the $a_i$'s, because it is the profinite completion of the topological fundamental group. ...
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What group is $\langle a,b \,| \, a^2=b^2 \rangle$?
In teaching my algebraic topology class, this group showed up as part of an easy fundamental group computation: $\langle a,b\mid a^2=b^2\rangle$. My first instinct was that this must be $\mathbb{Z}*\...
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2
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Lie Algebras and Simple Connectivity for general algebraic groups
In the representation theory of Lie groups (say, over $\mathbb{R}$ or $\mathbb{C}$), one can show that a Lie algebra homomorphism between the Lie algebras of two algebraic groups $G$ and $H$ always ...
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Inverting infinitely many points on an algebraic curve
This question is very naive, but that's why I'm asking it.
Say we begin with $\mathbb{A}^1_{\mathbb{C}}$. Let $U$ be the open disc around $0$ of radius $1$. Now invert all the $a$'s not in $U$: $Spec(...
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