Questions tagged [fundamental-group]
The fundamental-group tag has no usage guidance.
69 questions with no upvoted or accepted answers
2
votes
0
answers
317
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
$$
\...
- 14.2k
2
votes
0
answers
880
views
Question about the specialization map for Etale Fundamental Groups
Let $A$ be a complete, discrete valuation ring, and let $s$ (resp $\eta$) be the special (resp. generic) point of $S=Spec(A)$. Let $\phi:X \rightarrow S$ be a proper morphism and fix geometric base ...
- 133
1
vote
0
answers
85
views
Companions for positive characteristic arithmetic representations viewed as representations of the topological fundamental group?
Suppose $X / K$ is a variety over a finitely generated field over $\mathbb{Q}$. Fix an embedding $K \subset \mathbb{C}$ and let $\pi := \pi_1(X(\mathbb{C}), x)$ be the topological fundamental group. ...
- 3,730
1
vote
0
answers
98
views
Does there exist a simply connected surface with CM whose cotangent bundle is ample?
Does there exist a smooth projective complex surface $X$ such that,
(1) $\pi_1(X) = 0$
(2) $\Omega_X^1$ is ample
(3) the Mumford-Tate group of $H^2(X)$ is a torus
There exist examples with any two of ...
- 3,730
1
vote
0
answers
182
views
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$ ...
- 1,182
1
vote
0
answers
81
views
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 ...
- 679
1
vote
0
answers
238
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}$?
- 333
1
vote
0
answers
60
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{...
- 137
1
vote
0
answers
68
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 ...
- 1,593
1
vote
0
answers
175
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 ...
- 311
1
vote
0
answers
61
views
Restriction of irreducible integrable connections
Let $X$ be a complex analytic manifold and let $\mathcal{M}$ be an integrable connection on $X$, i.e. a $\mathcal{D}_X$-module which happens to be coherent as an $\mathcal{O}_X$-module. If $U$ is any ...
1
vote
0
answers
176
views
which sections of elliptic curves are conjugate?
Suppose you have a relative elliptic curves $f : E\rightarrow S$ (say $S$ is connected). Then suppose you have two sections $g,g' : S\rightarrow E$, corresponding to two sections $g_*,g'_*$ to the map ...
- 10.7k
1
vote
0
answers
136
views
algebraic varieties whose fundamental group is subgroup separable wrt subvariety subgroups
Call an algebraic variety $\pi_1$-subgroup separable iff,
for every $Y\subseteq X$ a closed subvariety and $\hat Y\xrightarrow{i} Y$ a normalisation of $Y$,
and subgroup $\Gamma=Im(\pi_1(\hat Y,y)\...
- 1,299
1
vote
0
answers
194
views
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(...
- 6,348
0
votes
0
answers
339
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:
...
- 29
0
votes
0
answers
286
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 ...
- 6,982
0
votes
0
answers
392
views
Galois cover corresponding to finite quotient of the étale fundamental group
Let $X$ be a connected scheme,$\pi_1(X,\bar{x})$ its étale fundamental group for some geometric point $\bar{x} : Spec(K) \rightarrow X$
and $E = \pi_1(X,\bar{x})/N$ a finite quotient of $\pi_1(X,\bar{...
- 181
0
votes
0
answers
147
views
A simple fundamental group of an hypersurface
Is there an example of analytic hypersurface in $C^n$ such that its fundamental group is simple i.e. does not have normal subgroups except the trivial group and the group itself ?
Thank you
EDIT : ...
0
votes
0
answers
173
views
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 $x \in X$. ...
- 952