38
votes
Accepted
"Cute" applications of the étale fundamental group
Using the étale fundamental group one can construct an injective group homomorphism
$\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \hookrightarrow \operatorname{Out}(\widehat{F_2})$
which is ...
- 286
32
votes
Is there a relationship between a quotient group of the fundamental group of X and the fundamental group of a quotient topology of X?
This is not always possible. (This answer was worked out in collaboration with Raymond Cheng.)
Example. Let $X$ be the pseudocircle: it is the finite quotient of the unit circle $S^1 \subseteq \...
- 28.7k
27
votes
Accepted
Does anyone know a basepoint-free construction of universal covers?
I think that homotopy-theorists often fall into the habit of working mainly with based spaces, even when they don't need to. It can be instructive to notice when the use of a basepoint is unnecessary, ...
- 55.9k
24
votes
Accepted
Does the isomorphic of the fundamental groups imply the existence of a mapping inducing an isomorphism?
No and no. For an explicit counterexample to 1. (which is also a counterexample to 2.) take the map $\mathbb{R}P^2\to \mathbb{R}P^{\infty}$.
- 10.8k
19
votes
3-manifold with fundamental group $\mathbb Z$
No. For example, take a copy of $S^1 \times S^2$ and remove the interior of a closed, nicely embedded, three-ball.
You will need to add the hypothesis of irreducibility (to rule out "punctures&...
- 28.1k
15
votes
Accepted
Simply connected slices
Yes, I think so. Let's show that every compact set $K\subset \Omega$ is contained in some compact contractible subset of $\Omega$. We use the fact that in a simply connected open subset of the plane ...
- 55.9k
14
votes
Accepted
A complex variety with a finite non-abelian simple fundamental group
Yes. In fact, Serre proved that any finite group is the fundamental group of a smooth projective complex variety. See Proposition 15 of:
J.-P. Serre, Sur la topologie des variétés algébriques en ...
- 44.8k
14
votes
"Cute" applications of the étale fundamental group
I think a fantastic application of the étale fundamental group is in extensions of the Chabauty-Coleman(-Kim) method. The original idea was that we could bound the rational points on a curve $C$ by ...
- 7,746
14
votes
For which spaces $S^n$ ($n\geq 2$) is a universal covering space?
The answer is quite complicated. To begin with, the universal cover of your space $X$ is a sphere $S^n$ with a free action of a finite group $G=\pi_1(X)$. The group $G$ has to have periodic cohomology....
- 16.2k
13
votes
Accepted
Surface bundles associated to a short exact sequence of groups
(1) It is not true that these groups are precisely the fundamental groups of $S$-bundles. The correct statement is that these groups are precisely the fundamental groups of $S$-bundles over a base ...
- 146
13
votes
Accepted
Representation theory of higher homotopy groups
There are many results that generalize the Riemann–Hilbert correspondence from the fundamental groupoid to the fundamental ∞-groupoid, for example:
Jonathan Block, Aaron Smith. A Riemann–Hilbert ...
- 37.8k
12
votes
Accepted
Fundamental groups of noncompact surfaces
In case anyone is interested, I wrote up a detailed account synthesizing the various answers here and correcting some issues I ran into. It is entitled "Spines of manifolds and the freeness of ...
- 44.8k
12
votes
Accepted
Are "large enough" finite etale covers arithmetic?
Let's assume that $X$ admits a $K$-point $x$ and use the corresponding geometric point as the base point. The existence of a rational point is in fact necessary for a positive answer, as explained by ...
- 7,377
12
votes
For which spaces $S^n$ ($n\geq 2$) is a universal covering space?
A nice and quick survey on the groups acting freely on the sphere is given in chapter 3 of https://webusers.imj-prg.fr/~bernhard.keller/ictp2006/lecturenotes/skowronski.pdf .
Theorem 3.26 gives a nice ...
- 26.5k
12
votes
Accepted
Example of three dimensional atoroidal Poincaré duality group with some pathology
One answer to your question comes from the paper The Weber-Seifert dodecahedral space is non-Haken by Burton, Rubinstein, and Tillmann.
An earlier example is (say) the $(1, 2)$-Dehn filling of the ...
- 28.1k
11
votes
Fundamental groups of noncompact surfaces
Related to Mohan's answer, one can give an overkill proof using the fact that non-compact Riemann surfaces are Stein, and every complex $n$-dimensional Stein manifold is homotopy equivalent to an n-...
- 68.8k
11
votes
Accepted
Fundamental groups of complements of divisors in $\mathbb P^2$
I'd leave this as a comment, but I don't have enough reputation. Consider the long exact sequence in homology of the pair $(\mathbb{P}^2, \mathbb{P}^2-D)$. Since $H_1(\mathbb{P}^2,\mathbb{Z}) = 0$ and ...
- 666
11
votes
Is $\mathbb{R}^3 \setminus \mathbb{Q}^3$ simply connected?
$\newcommand\R{\mathbb{R}} \newcommand\cH{\mathcal{H}} \newcommand\bbD{\mathbb{D}}$
I'm marking this community wiki since it's not so much an answer but an attempt to make it clear to the casual ...
11
votes
Accepted
Do complex varieties have a dense open subset with residually finite fundamental group?
(I'm converting my comment to an answer.)
In SGA4 exp XI, Artin constructs a nonempty Zariski open $U\subset S$ which admits a sequence $U=U_n \to U_{n-1}\to\ldots $ which are topological fibrations ...
- 35.2k
11
votes
Galois theory, topos vs fundamental groups
I'm not sure I understand what you precisely want to know, so I'll try to clarify a few things from the topos theoretic point of view, which I hope will answer your questions:
Relation to Galois ...
- 42.4k
11
votes
Can one compute the fundamental group of a complex variety? Other topological invariants?
Suppose that $X\subseteq\mathbb{C}P^n$ is a smooth variety. There are various ways to define families of smooth maps $\mathbb{C}P^n\to\mathbb{R}$; for example, we can choose a line $L_0<\mathbb{C}^...
- 56.9k
11
votes
Can one compute the fundamental group of a complex variety? Other topological invariants?
There are algorithms to compute triangulations of real algebraic varieties, and thus of complex algebraic varieties. Sadly, the algorithms tend to be doubly exponential in the size of the input data. ...
- 96.4k
11
votes
Does anyone know a basepoint-free construction of universal covers?
Part of 10.5.8 of Topology and Groupoids is, in a more usual notation, essentially the following, in which $\sigma, \tau$ are the source and target maps, $St_G x$ is $\sigma ^{-1} x$, by $N$ is ...
- 12.3k
11
votes
Accepted
Fundamental group of the smooth locus of a normal algebraic surface is a quotient of that of a Zariski open subset
The question is essentially asking why the inclusion of $Z$ into the smooth locus $Y$ induces a surjection on $\pi_1$. The ambient space $X$ is not really relevant here. You're removing some Zariski ...
- 8,803
10
votes
Accepted
Mapping class group and representation of fundamental group of Riemann surfaces
There are counterexamples as soon as $g > 1$.
Let $n$ be the number of surjective homomorphisms $\pi_1(S) \to A_5$, up to $S_5$-conjugacy. (We can see that $n \geq 1$ using the fact that $A_5$ ...
- 148k
10
votes
Accepted
Is $H_{et}^1(X,F) = H^1(\pi_1^{et}(X), F(\bar{k}))$ true?
This is true, yes. More generally, if $X$ is a scheme and $F$ is a locally constant étale sheaf of finite abelian groups on $X$, then
$$
H^1_{et}(X,F) = H^1(\Pi_1^{et}(X), \tilde F),
$$
where $\Pi_1^{...
- 8,972
10
votes
Accepted
Can we define fundamental groups functorially for non-pointed path connected topological spaces?
For any such lift $\widetilde{FG}:\mathrm{pTop}\to \mathrm{Gp}$ the induced map $pTop(X,Y)\to Gp(\widetilde{FG}(X),\widetilde{FG}(Y))$ must factor through the set of homotopy classes of the maps ...
- 7,377
10
votes
Are "large enough" finite etale covers arithmetic?
Here's a simple argument assuming $X$ admits a $K$-rational point, and that $X$ has a finitely generated geometric fundamental group. In fact the "further" covering $X''$ can be chosen to be ...
- 10.7k
10
votes
Are "large enough" finite etale covers arithmetic?
Adding on Will's and Sasha's answers, the condition of having a rational point, or at least a "1-truncated homotopy fixed point" for the action is necessary. For example, let $C_2$ act on ...
- 4,189
9
votes
Accepted
Fundamental groups of non-orientable closed four-manifolds
Let $G$ be a finitely presented group
$$ G = \langle g_1, g_2, \cdots, g_n | R_1, R_2, \cdots, R_m \rangle$$
One standard way to realize this as the fundamental group of a compact $4$-manifold is to ...
- 44.3k
Only top scored, non community-wiki answers of a minimum length are eligible