All Questions
Tagged with fundamental-group gt.geometric-topology
22 questions
2
votes
1
answer
287
views
How does hyperelliptic involution act on the standard generators of the fundamental group of surfaces of genus g with n punctures?
Let $S_{g,n}$ be the surface of genus $g$ with $n$ punctures. We know that $\pi_1(S_{g,n})$ admits a presentation:
$$\left\langle~ \alpha_1,\beta_1,\dots, \alpha_{g},\beta_{g},\gamma_{1},\dots,\gamma_{...
- 79
1
vote
1
answer
279
views
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 ...
- 214
7
votes
2
answers
367
views
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,...
- 1,177
7
votes
1
answer
291
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\...
- 18.7k
2
votes
1
answer
282
views
Lifting of a proper map in the cover is a proper map
Let $M$ be an orientable surface without boundary$($I am not assuming $M$ is compact, it can be non-compact$)$. Let $\Phi: M\to M$ be a proper homotopy-equivalnce$($A proper homotopy-equivalence can ...
- 265
5
votes
2
answers
916
views
3-manifold with fundamental group $\mathbb Z$
Let $M$ be a compact $3$-manifold with nonempty boundary. If $\pi_1(M)=\mathbb Z$, can we prove that $M$ is homeomorphic to $S^1 \times D^2$?
- 2,535
8
votes
1
answer
575
views
Understanding fundamental group of Poincare homology sphere
I'm currently reading Knots, Links, Braids, and 3-Manifolds by V. V. Prasolov and A. B. Sossinsky. I have trouble understanding the following picture. The dashed line denotes a trefoil whose tubular ...
- 103
4
votes
0
answers
397
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 ...
- 976
1
vote
2
answers
353
views
Non-self-intersecting paths on $\mathbb{C}\setminus\{0,1\}$ [closed]
Let us make two small holes around points $0$ and $1$ on the complex plane and consider non-self-intersecting paths that start on the boundary of one hole and finish at the boundary of the another. It ...
- 29
1
vote
1
answer
327
views
Fundamental group of the complement of cell subcomplexes
Given a regular CW complex stucture on a manifold $C$ of dimension $n$ and a subcomplex $D$ of dimension $n-2$, I want to compute the fundamental group of the complement $\pi_1(C\setminus D)$. A ...
- 154
4
votes
1
answer
270
views
Invariant lifts of a closed curve on a surface of genus > 1
I am learning some things about surfaces of genus greater than $1$, and I am trying to answer this question :
Let $S$ be a compact and orientable surface of genus $g \geq 2$, and $c$ a closed curve ...
- 41
6
votes
1
answer
237
views
Example similar to the Griffiths twin cone but with fundamental group that allows surjection onto $\mathbb Z$
The Griffiths twin cone is an example of a wedge sum of two contractible spaces being non-contractible. Namely, it is the wedge sum $\mathbb G=C\mathbb H\vee_p C\mathbb H$ of two coni over the ...
17
votes
1
answer
574
views
Simply connected slices
Assume $\Omega$ is an open set in $\mathbb R^3$
such that the intersection of $\Omega$ with any horizontal plane is simply connected.
Can you prove that $\Omega$ is simply connected?
(Note that ...
- 45k
12
votes
1
answer
798
views
Finite covers of hyperbolic surfaces and the `second systole´
We are interested in the following ´relative´ version of residual finiteness for fundamental groups of surfaces. Similar discussions where given in this question: injectivity radius of hyperbolic ...
- 3,928
2
votes
1
answer
125
views
Images of boundary surfaces in 3-manifold groups
Let $M$ be a compact connected 3-manifold and let $S$ be a closed connected surface in $\partial M$. Let $G$ be the image of the map $\pi_1(S) \to \pi_1(M)$ induced by inclusion. I was reading the ...
- 5,349
6
votes
1
answer
405
views
Fundamental groups of hyperbolic $4$-manifolds and $\rm CAT(0)$ cube complexes
Suppose $M^4$ is a compact hyperbolic (i.e. curvature $-1$) $4$-manifold and $\Gamma\cong\pi_1(M^4)$.
Is there any expectation whether $\Gamma$ acts properly and co-compactly on a $\rm CAT(0)$ cube ...
- 14.3k
5
votes
1
answer
1k
views
On the fundamental group of closed 3-manifolds
I know that every finitely presented group can be realized as the fundamental group of a compact, connected, smooth manifold of dimension 4 (or higher). In dimension 2 there are strong restriction on ...
- 683
4
votes
2
answers
583
views
a question on rank of fundamental group
Assume $G$ be the fundamental group of a closed orientable hyperbolic 3-manifold. Let
$G_{1} = \langle a_{1},...,a_{k} \rangle$ be a free subgroup of $G$, and let $G_{2}=\langle a_{k+1} \rangle$ be a ...
- 841
3
votes
3
answers
352
views
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 ...
- 215
62
votes
9
answers
9k
views
Fundamental groups of noncompact surfaces
I got fantastic answers to my previous question (about modern references for the fact that surfaces can be triangulated), so I thought I'd ask a related question. A basic fact about surface topology ...
- 44.8k
24
votes
4
answers
4k
views
Fundamental group of 3-manifold with boundary
Is it true that any finitely presented group can be realized as fundamental group of compact 3-manifold with boundary?
- 45k
73
votes
10
answers
22k
views
Galois groups vs. fundamental groups
In a recent blog post Terry Tao mentions in passing that:
"Class groups...are arithmetic analogues of the (abelianised) fundamental groups in topology, with Galois groups serving as the analogue ...
- 2,806