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Results tagged with gt.geometric-topology
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user 1463
Topology of cell complexes and manifolds, classification of manifolds (e.g. smoothing, surgery), low dimensional topology (e.g. knot theory, invariants of 4-manifolds), embedding theory, combinatorial and PL topology, geometric group theory, infinite dimensional topology (e.g. Hilbert cube manifolds, theory of retracts).
2
votes
Fundamental group of the complement of a codimension two submanifold
As you suspect, in dimension 3 the answer to your first question should be "no". Indeed, if one deletes any collection of circles from $M$ then the complement is a 3-manifold with toroidal boundary. S …
- 25.2k
8
votes
Finite two-relator groups and quotients of knot groups
Question 1:
As mentioned in comments, presentations with the same number of generators and relations are called balanced. The triviality problem for balanced presentations appears to be a question of …
- 25.2k
10
votes
Fundamental group of a generalized connected sum
This basic question is unfortunately not well explained anywhere in the literature that I know of, although the answer is well known to lots of people. When $\pi_1(S)$ embeds into $\pi_1(M)$ and $\pi_ …
- 25.2k
7
votes
Accepted
Existence of a surface group ensures the existence of a $\pi_1$-injective immersed surface
This fact doesn’t need $M$ to be hyperbolic. It just needs one general theorem about 3-manifold topology, namely the Scott core theorem.
Let $N\to M$ be the covering space corresponding to the subgrou …
- 25.2k
5
votes
Residual finiteness of hyperbolic 3-manifold groups
Sam Nead's answer does it, but perhaps I can offer a slightly different perspective on Question 1. No complicated hyperbolic gluing results are needed.
I assume we are satisfied with the characterisat …
- 25.2k
4
votes
Elevator pitch for the Virtual Fibering Theorem
The Virtual Fibring theorem provides the topological classification of closed 3-manifolds.
Very roughly, after passing to finite covers, we can build 3-manifolds by gluing together simpler pieces that …
- 4,713
6
votes
Accepted
Order of a loop around a cone point
As mentioned in comments, the answer is "yes" and there are many ways to see it. I would refer you to §2 of Peter Scott's survey paper "The geometries of 3-manifolds", in which he discusses 2-dimensio …
- 25.2k
9
votes
Accepted
Is every automorphism of $\mathrm{Aut}^+(F_2)$ induced by conjugation inside $\mathrm{Aut}(F...
$\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Out{Out}$If I have chased through the literature correctly, I think the answer to your question is "yes". Specifically:
Dyer–Formanek–Grossman showed …
- 12.9k
9
votes
Amalgamated product acting on CAT(0) cube complex
To extend the gluing result from Bridson--Haefliger to non-positively curved cube complexes, it is important to work in the correct category.
If we want the result to also be a non-positively curved c …
- 25.2k
2
votes
Identifying a curve on a closed surface of genus 4
As mentioned in comments, your picture is not entirely accurate. But perhaps this is what you're looking for?
(Note that, if you had chosen a different gluing pattern for your once-punctured genus-tw …
- 25.2k
7
votes
Generate $\mathrm{Mod}(S_g)$ by two Dehn twists
This is addressed in §3.5.2 of Farb and Margalit's Primer on Mapping Class Groups. The subgroups of mapping class groups generated by two Dehn twists $T_a,T_b$ are one of:
$\mathbb{Z}$ if the curves …
- 25.2k
4
votes
Accepted
Examples of finite polyhedra with finitely generated simple fundamental group
As suggested in the comments, what you are asking for is essentially the presentation complex of a finitely presented, infinite, simple group. Thus it suffices to exhibit a presentation for such a gro …
- 25.2k
5
votes
HNN decomposition of finite rank free group over infinite rank subgroups
You haven't quite given the full strength of Swarup's result, which is what makes it so useful: moreover, $H_2$ and $tHt^{-1}$ are conjugate into $J_2$. The geometric interpretation is that a natural …
- 25.2k
5
votes
Accepted
Does length spectrum determine a hyperbolic 3-manifold? What if we also know holonomies?
I think your questions are answered by this paper of Leininger, McReynolds, Neumann and Reid. In their terminology, your first question is asking about manifolds with equal length sets, and your seco …
- 25.2k
3
votes
From topological actions on $\mathbb{R}^3$ to isometric actions
My earlier attempt at an answer was a bit of a mess -- let me have another go.
The hypotheses of the question introduce several technical difficulties, but I'm unsure which are crucial and which can b …
- 25.2k