Questions tagged [fundamental-group]

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Etale fundamental group of a curve in characteristic $p$

Let $C$ be a connected, smooth, proper curve of genus $g$ over an algebraically closed field $k$ of characteristic $p>0$. Let $\pi_1(C)$ be the etale fundamental group of $C$ - I only care about ...
jacob's user avatar
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On a homological finiteness condition

Assumption: $X$ is a connected CW complex, and $H_{\ast}(X;\mathbb{Z})=\bigoplus_{n \geq 0} H_n (X; \mathbb{Z})$ is finitely generated. Question: does there exist a finite CW complex $Y$ and a map $f:...
Johannes Ebert's user avatar
16 votes
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732 views

What would be the simplest analog of Langlands in algebraic topology?

It is oversimplified, I know, but just as a superficial analogy, one may think of the fact that abelianization of the fundamental group is the first homology group, as some remote relative of class ...
მამუკა ჯიბლაძე's user avatar
16 votes
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581 views

Are there any examples of hyperbolic curves over finite fields such that the action of frobenius on its prime-to-$p$ fundamental group is known?

Let $X$ smooth curve over a finite field $\mathbb{F}_q$ of type $(g,n)$ - that is, $X$ is an open subscheme of its genus $g$ compactification obtained by removing $n$ points. Any such curve ...
Will Chen's user avatar
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16 votes
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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 \...
Hiro Lee Tanaka's user avatar
13 votes
0 answers
847 views

About maps inducing bijections on homotopy classes

Let us assume that $f:X \to Y$ is a map of connected CW complexes, having the following property: if $K$ is a finite CW complex, then the induced map $f_{\ast}:[K,X] \to [K,Y]$ on \emph{free} homotopy ...
Johannes Ebert's user avatar
10 votes
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232 views

On tangent space to the fundamental group scheme

Let $X$ be a smooth, projective complex curve of genus at least $2$. If I understand correctly, after choosing a base point, one can associate to $X$, a fundamental group scheme $\pi$. I am trying to ...
Ron's user avatar
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427 views

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 ...
o a's user avatar
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8 votes
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288 views

Relationships among constructions of fundamental group for schemes

There seem to be several constructions of fundamental group for schemes and stacks: by Grothendieck, Deligne, Nori, Noohi, Esnault-Hai, Vakil-Wickelgren, perhaps others as well. I am trying to ...
Galois groupie's user avatar
8 votes
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253 views

Fundamental group of moduli space of K3's

According to Rizov (https://arxiv.org/abs/math/0506120), the moduli stack of primitively polarized K3 surfaces of degree 2d $\mathcal{M}_{d}$ is a Deligne-Mumford stack over $\mathbb{Z}$. I'm looking ...
CKlevdal's user avatar
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$p$-adic representations of the fundamental group of a smooth proper curve over a finite field

This question is very general. Let $C$ be a smooth and proper curve over a finite field ${\bf F}_p$. Are there any general results or conjectures on continuous non abelian representations $$ \pi_1(C)\...
Damian Rössler's user avatar
8 votes
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219 views

Does the fundamental group identify group structure on subvarieties of products of curves?

Let $C_1,\dots, C_n$ be smooth curves over $\overline{\mathbb F}_p$, not necessarily proper. Let $X$ be a subvariety of $C_1 \times \dots \times C_n$. I'm interested in the natural map: $$ \pi_1^{ab}(...
Will Sawin's user avatar
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7 votes
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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 ...
Benedikt's user avatar
6 votes
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218 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\...
John Pardon's user avatar
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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 $\...
Antoine Ducros's user avatar
6 votes
0 answers
420 views

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 $\mathbb{Z}$, $\mathbb{F}_q$...
genshin's user avatar
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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 ...
Jake Levinson's user avatar
5 votes
0 answers
388 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 ...
user avatar
5 votes
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Nonabelian Hodge structure for noncompact curves and Hodge structure on the fundamental group

Nonabelian Hodge theory, introduced by C. Simpson and others, may be interpreted as a description of the (real) Hodge structure on the fundamental group (say, of a compact curve) in terms of some ...
nikitamarkarian's user avatar
5 votes
0 answers
198 views

Algebraic fundamental group without regularity at infinity

Suppose $X$ is a smooth (connected) variety over $\mathbb{C}$. Let $\mathscr{C}$ be the category of finite rank vector bundles on $X$ equipped with an integrable connection, and let $\mathscr{C}'$ be ...
Julian Rosen's user avatar
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278 views

Nori fundamental group and etale fundamental group in positive characteristic

Let $X$ be a smooth projective surface over an algebraically closed field of char $p > 0$. Suppose that $\pi_{1}^{et}(X) = \{1\}$. Can Nori fundamental group scheme of $X$ be non-trivial?
AlekseiG's user avatar
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Fundamental groups of Hirzebruch's line arrangement varities

Let $\Lambda$ be a line arrangement in $\mathbb{P}^2$ and $n > 0$ an integer. Then Hirzebruch defined a smooth projective surface $H(\Lambda, n)$ as the minimal desingularization of a covering $Y \...
Ben C's user avatar
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4 votes
0 answers
121 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^{...
Jacques's user avatar
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0 answers
58 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+...
Samuele's user avatar
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4 votes
0 answers
362 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 ...
Martin Tancer's user avatar
4 votes
0 answers
193 views

Geometric fundamental group and algebraically closed residue field

my questions relates to the following talk of Tsuji: https://www.youtube.com/watch?v=2brDj26phP0 At around 10:30 of the video, Tsuji is interrupted by a man stating that his construction does not ...
Konstantin's user avatar
4 votes
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112 views

Classification of manifolds with ${\rm Ric}\geq 0$ wrt fundamental group

Note that $n$-manifolds $M$ with ${\rm Ric}\geq 0$ has a fundamental group of polynomial growth of degree $\leq n$ (proof : use Bishop volume theorem). (Here a group $\Gamma$ is said to have ...
Hee Kwon Lee's user avatar
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4 votes
0 answers
204 views

How do we see the rank of the braid group?

The only presentation of the braid group that most people ever see is the standard Artin presentation $$B_n=\langle σ_1,\cdots,σ_{n−1}|\ σ_iσ_j=σ_jσ_i\ \ (|i−j|>1),\ σ_iσ_{i+1}σ_i=σ_{i+1}σ_i σ_{i+...
dvitek's user avatar
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4 votes
0 answers
490 views

Exact sequence of the fundamental group of the general fiber

Let $f\colon X\rightarrow Y$ be a morphism of complex algebraic varieties. Let $y\in Y$ be a general point, then we have a sequence of homomorphisms of fundamental groups induced by the inclusion of ...
Joaquín Moraga's user avatar
4 votes
0 answers
150 views

local systems with cyclic monodromy

In their book "Lectures on vanishing theorems", Esnault and Viehweg used finite cyclic covering of varieties constructed as follows: Let $X$ be a smooth projective variety over some field $k$ of ...
cyc83's user avatar
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3 votes
0 answers
99 views

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 ...
Piotr's user avatar
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3 votes
0 answers
185 views

What should be unipotent de Rham homotopy group?

What exactly should unipotent $\pi_1^\text{dR}$ be conceptually? What formal properties should it satisfy? This seems to be answered by Chen's theorem, which is stated in Corollary 3.269 of Multiple ...
W. Zhan's user avatar
  • 343
3 votes
0 answers
91 views

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 ...
oleout's user avatar
  • 865
3 votes
0 answers
56 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 $L$ ...
Faniel's user avatar
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206 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\...
Chetan Vuppulury's user avatar
3 votes
0 answers
255 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 ...
user267839's user avatar
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3 votes
0 answers
228 views

Is there a reasonable notion of universal cover for schemes over arbitrary fields?

Let $X$ be smooth projective variety, such as an elliptic curve. We know that $X$ does not have a universal cover in the category of schemes. However, if $k = \mathbf{C}$, then $\tilde{X}$ exists as ...
Kim's user avatar
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3 votes
0 answers
243 views

First thoughts about fundamental group of a topological (Lie) groupoid

I am reading the paper Chern-Weil map for principal bundles over groupoids. In page number $13$, authors say let us recall the definition of fundamental group of a topological groupoid. But, they ...
Praphulla Koushik's user avatar
3 votes
0 answers
161 views

Recover an etale fundamental group from local fundamental groups?

Suppose we have a scheme $S$ and an etale covering $\{ U_i \to S \}$. How much information about $\pi_1^{et} (S)$ can we recover from all the $\pi_1^{et} (U_i)$ ? Are there special conditions we can ...
JJJ's user avatar
  • 167
3 votes
0 answers
135 views

How does the fundamental group of $\mathbb G_{m,S}$ depend on the base scheme

Let $S$ be an integral noetherian regular scheme, and let $X =\mathbb{G}_{m,S}$. How to compute $\pi_1^{et}(X)$? Note. I am only interested in the part not coming trivially from the finite etale ...
Srsly's user avatar
  • 39
3 votes
0 answers
595 views

Monodromy representations are "quasi-unipotent"

Let $S$ be a smooth complex algebraic variety, let $b$ be a closed point of $X$, and let $f:X\to S$ be a polarized family of smooth projective varieties over $S$. This induces a monodromy ...
Randy's user avatar
  • 113
2 votes
0 answers
81 views

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 (...
Manenky's user avatar
  • 21
2 votes
0 answers
106 views

Is the connecting map $\pi_2(B) \to \pi_1(F)$ ever nonzero in smooth proper families?

Suppose that $X, B$ are smooth irreducible varieties over $\mathbb{C}$ and $f : X \to B$ is a smooth proper morphism. Then we can consider the homotopy exact sequence: $$ \pi_2(B) \to \pi_1(F) \to \...
Ben C's user avatar
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2 votes
0 answers
132 views

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 ...
Ben C's user avatar
  • 3,281
2 votes
0 answers
85 views

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 ...
oleout's user avatar
  • 865
2 votes
0 answers
171 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 ...
user127776's user avatar
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2 votes
0 answers
159 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*} ...
Aoi Koshigaya's user avatar
2 votes
0 answers
242 views

How to compute fundamental groups of slice disk complements?

To compute the fundamental group of the complement $S^3 \setminus K$ of a knot, one usually uses the Wirtinger algorithm. Is there a similarly well-established procedure for computing the fundamental ...
Levi Ryffel's user avatar
2 votes
0 answers
92 views

Realizing an amalgamated product of groups by splitting a closed manifold along a codimension 1 submanifold

In the paper "A splitting theorem for manifolds" by S.E. Cappell, https://www.maths.ed.ac.uk/~v1ranick/papers/capsplit.pdf the following "inverse" of the Seifert-van Kampen theorem for closed ...
user147418's user avatar
2 votes
0 answers
293 views

A homomorphism in the long exact sequence of a fibration for a homogeneous space of a Lie group

Let $G$ be a connected Lie group, and let $H\subset G$ be a (closed) Lie subgroup, not necessarily connected. Set $X=G/H$. The fibration $j\colon G\to X$ with fiber $H$ induces an exact sequence $$ \...
Mikhail Borovoi's user avatar