Questions tagged [gt.geometric-topology]
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).
13 questions from the last 30 days
3
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0
answers
143
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Cell structure on the function space $\operatorname{Hom}(X,Y)$
By the Theorem of Milnor in his paper "On spaces having the homotopy type of a CW-complex", the function space $\operatorname{Hom}(X,Y)$ (with the compact-open topology) is homotopy ...
- 140
7
votes
1
answer
210
views
Preserving non-conjugacy of loxodromic isometries in a Dehn filling
Suppose that $g$ and $h$ are non-conjugate loxodromic isometries in a cusped hyperbolic $3$-manifold $M$ of finite volume. Fix a cusp $T$ of $M$. Can I choose a hyperbolic Dehn filling of $M$ along $...
-1
votes
0
answers
114
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Stability of flow map
$\DeclareMathOperator\Diff{Diff}$Setting:
Let $(M,g)$ be a compact and connected $C^{\infty}$-Riemannian manifold. Let $d_g$ denote the induced shorted path metric and equip $C^{\infty}(M)$ with the ...
- 5,405
12
votes
1
answer
557
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+200
Fundamental group of the complement of a codimension two submanifold
Let $M$ denote an arbitrary smooth, closed, connected, n-dimensional manifold for $n\geq 4$. For every such $M$, does there exist a closed (not necessarily connected!) codimension two submanifold $S \...
- 1,174
3
votes
1
answer
157
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Geometry and topology of Fuchsian character varieties
Consider the hyperbolic space, $\mathbb H^2$. A Fuchsian group is a discrete subgroup of $\text{PSL}(2,\mathbb R)$. We can generate tessellations, especially $\{p,q\} \;\text{tesellations}$ of $\...
- 357
3
votes
1
answer
269
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$\mathrm{SL}(2,\mathbb{Z})$ finitely generated by using the Schwarz-Milnor lemma
Recently, I have been studying the modular group $G=\mathrm{SL}(2,\mathbb{Z})$, and I am trying to prove $G$ is finitely generated by using the Schwarz-Milnor lemma in geometric group theory.I am ...
- 111
7
votes
1
answer
177
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Ergodicity of action of finite index subgroups in the boundary
Let $\Gamma < \operatorname{PSL}_2(\mathbb{R})= \text{Isom}^+(\mathbb{H^2})$ be a discrete subgroup. Suppose $\Gamma$ acts ergodically on the boundary of the hyperbolic plane $\partial{\mathbb{H}^2}...
- 1,101
4
votes
1
answer
153
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Kirby diagram of the complement of a subhandlebody of a smooth closed 4-manifold
Let $X$ be a smooth closed connected 4-manifold. It admits a handlebody structure, having a unique 0- and a unique 4-handle. We can express the handlebody structure as a Kirby diagram (https://en....
- 213
10
votes
1
answer
659
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Are there any tests for knowing whether a topological space admits a CW structure?
We know that for n $\ge$ 5, a manifold admits a piecewise linear structure if and only if its Kirby-Siebenmann class vanish and Galewski and Stern showed the existence of a similar invariant to test ...
- 168
3
votes
1
answer
132
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Is a simply connected locally 2-connected complex a union of spheres and planes?
Let $X$ be a (potentially infinite) 2-dimensional simplicial complex. Then each link at a vertex $x\in X$ is a graph.
Question. If $X$ is simply connected and each link is 2-connected (in the sense ...
- 13.6k
1
vote
0
answers
69
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"Bad" valid edge contractions
In this paper, an edge contraction of a simplicial complex $\Gamma$ is defined as the operation of removing the neighborhood $N_e\Gamma$ of the edge $e=\{0,1\}$ and identifying $N_0\partial N_e\Gamma$ ...
- 11
2
votes
1
answer
380
views
Is it always possible to connect the endpoints of a smooth injective path, so the resulting path is smooth, closed and simple?
Motivation
The motivation for this question arises from the following problem: Is there a closed, simple, and infinitely differentiable path on the (complex) plane such that every straight line has a ...
12
votes
1
answer
323
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Does every mapping class group embed into some $\mathrm{Out}(F_n)$?
The title is pretty much the whole question. Let $S_g$ be a closed, oriented surface of genus $g$. Does there exist $n$ such that the mapping class group $\mathrm{Mod}(S_g)$ embeds as a subgroup of $\...
- 3,740