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).
4,259 questions
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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 ...
- 12.9k
7
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1
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218
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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 $...
- 68.9k
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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
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115
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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
3
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1
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158
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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 $\...
- 357
3
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1
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269
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$\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 ...
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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}...
- 12.3k
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1
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154
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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....
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10
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1
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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 ...
- 66.3k
3
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1
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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 ...
- 12.3k
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3
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3k
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The Jones polynomial at specific values of $t$
I've been calculating some Jones polynomials lately and I was just curious if there was a "physical" (or, rather, geometric) meaning to evaluating the Jones polynomial at a particular value of $t$.
...
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1
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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 ...
- 91.8k
1
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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 $N_0\partial N_e\Gamma$ ...
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470
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Cluster algebras of type A and X
I will base my question on Fock and Goncharov's paper Dual Teichmüller and lamination spaces.
Let $S$ be a surface with boundaries, marked points on such boundaries, punctures and boundaries without ...
CommunityBot
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12
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1
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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 $\...
- 63.9k
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2
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$\mathrm{GL}_n(\mathbb{Z}_2)=\mathrm{Out}(F_n)/\langle\langle \epsilon_1,\dots,\epsilon_n\rangle\rangle$
In a paper I found the following result:
$$\mathrm{GL}_n(\mathbb{Z}_2)=\mathrm{Out}(F_n)/\langle\langle \epsilon_1,\dots,\epsilon_n\rangle\rangle$$
However, they got the result as a corollary of a ...
- 7,893
3
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0
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119
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Signature vs commensurability
If a closed oriented $4n$-dimensional manifold $M$ has an orientation-reversing homeomorphism, then the signature $\sigma(M)$ of the intersection form vanishes. More generally, $\sigma(M) = 0$ if $M$ ...
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2
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3k
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The word problem in braid groups
I have read a statement from Sossinsky and Prasolov' s book "Knots, Lİnks, Braids and 3-Manifold", it says that two reduced word represent isotopic braids if and only if they have the same reduced ...
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3
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115
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Extending triangulations on surfaces
Suppose that $M$ is a surface (i.e., connected topological 2-manifold, I am willing to assume compact, but possibly with boundary), $K$ a finite simplicial complex, and $f$ an embedding of $K$ into $M$...
- 3,115
4
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177
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Basis of topology on space of properly embedded smooth manifolds
In A Short Exposition of the Madsen-Weiss Theorem, Hatcher discusses (starting at p.6) a basis for a topology on the space $\mathcal{C}^n$, the space of all smooth oriented $d$-dimensional ...
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2
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189
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Necessary condition for invertible knot concordance from both ends
It is clear that if $K_1$ and $K_2$ are two concordant knots by a concordance that only present ambient isotopic phenomena (no saddles, maxima, or minima) they are invertible concordant from both ends....
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1
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Can increasing the winding number of a 2-cell make a CW complex embeddable?
Let $X$ be a 2-dimensional CW complex and let $c\subset X$ be a 2-cell attached along a simple closed loop $\gamma$ in the 1-skeleton $X^{(1)}$.
For a natural number $n\ge 2$ consider the operation of ...
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1
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378
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Why is this Brieskorn manifold a rational homology sphere?
In Némethi's book "Normal surface singularities", Example 5.1.17, there is a formula to find the Seifert invariants of a Brieskorn complete intersection $\Sigma(a_1,...,a_n)$. I am ...
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1
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582
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Does every oriented $3$-dimensional submanifold of $\mathbb{R}^6$ bound an oriented $4$-dimensional submanifold?
In my recent research, I encountered the following problem about embeddings. Let $M^3$ be a closed compact oriented smooth $3$-dimensional submanifold of $\mathbb{R}^6$. Does there exist a compact ...
- 161
1
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1
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91
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When is a 2-bridge knot hyperbolic?
It is known that 2-bridge knots in $S^3$ can be classified by the Schubert form. My question is: which 2-bridge knots are hyperbolic? (Do we have a complete classification for hyperbolicity in 2-...
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2
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2k
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Is there a continuous partition of space into circles?
Question 1. Is there a continuous partition of space $\mathbb{R}^3$ into circles?
I strongly suspect not.
It is well-known by diverse arguments that space can be partitioned into circles. There is an ...
- 12.3k
5
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2
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199
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Handle decompositions subordinate to an open cover
Let $M$ be a compact smooth manifold and let $\{U_i\}_{i\in I}$ be an open cover.
We say a handle decomposition of $M$ is subordinate to the open cover if each handle is contained in a $U_i$. Do such ...
- 19.4k
4
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1
answer
236
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If a lattice can be embedded into $\mathbb Q^n,\langle-1\rangle^n$, then can it be embedded into $\mathbb Z^n,\langle -1 \rangle^n$?
Given a graph with negative integers on each vertex $\Gamma$ there is a corresponding intersection lattice denoted $Q_\Gamma$, a free $\mathbb Z$ module generated by the vertices, endowed with a ...
- 12.9k
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3
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When does a CW-complex of dimension 2 embed in $\Bbb R^4$?
Let $X$ be a finite CW-complex of dimension two having just one 0-cell
(+ finitely many 1-cells + finitely many 2-cells).
Is it true that X can be embedded in $\Bbb R^4$?
If true, is it due to ...
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1
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Existence of orientable finite volume complete cusp hyperbolic 3-manifolds $\mathbb{H}^3 / \Gamma$, where $\Gamma$ has no parabolic generators?
Let $K$ be a hyperbolic knot, i.e., $S^3 - K$ is an orientable finite volume cusp hyperbolic 3 manifold. Let $M=S^3 - K$ then $M= \mathbb{H}^3/\Gamma$, where $\Gamma$ (Kleinian group) is discret ...
- 12.3k
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0
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128
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The smallest dihedral angle of convex polyhedrons
Given a set of points $\{x_{k}\}_{k=0}^{m} \subset \mathbb{R}^n$, is it always possible to find a constant $c=c(m,n)>0$, depending only on the dimension $n$ and the number $m$, such that, after ...
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Is there a 2 component link with full symmetry?
If you take a (labelled, oriented) 2 component link, it has a symmetry group which is a subgroup of the 16 element group $Z2 \times (Z2 \times Z2 \ltimes S_2)$ (mirror, reverse each component, swap ...
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3
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51
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Asymptotic dimension of graph families representing each finite group
Frucht's theorem says every finite group is isomorphic to the automorphism group of a simple graph $G$ (with no loops, multiple edges or directed edges).
There has been interest in finding classes of ...
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1
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400
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Finitely generated groups with Hölder-exotic space of ends?
The space of ends of a finitely generated group is always homeomorphic to 0, 1, 2 points, or a Cantor set, and in which of these 4 cases it falls is governed by Stallings' characterization (wikipedia ...
- 63.9k
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1
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Is every Cantor set $C\subseteq\mathbb R^{\infty}$ the limit set of a Fuchsian group?
Is every Cantor set $C\subseteq\mathbb R^{\infty}$ the limit set of a Fuchsian group?
Let's recall the definitions. A Fuchsian group $G$ is a discrete subgroup of $\operatorname{PSL}(2,\mathbb R)=\...
- 63.9k
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0
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Is there a closed aspherical manifold with infinitely many symmetries and without essential immersed tori?
The precise question is the following:
Is there a closed aspherical manifold $M$ of dimension $n\geq 3$ such that Out($\pi_1(M)$) is infinite and $\pi_1(M)$ does not contain $\mathbb Z \times \mathbb ...
- 10.5k
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299
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Mapping class group of non-orientable three manifold
I do not work in topology, but for some reason we need to know the mapping class group of certain non-orientable 3 manifolds. We found answers online for a lot of orientable manifolds. But we still ...
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1
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150
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Slice-ribbon conjecture in other 3-manifolds
There is some notion of what it means for a knot $K\hookrightarrow M$ to be "slice." In particular, we may ask, for example, that there is a topologically embedded disk in $M\times[0,1]$ ...
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2
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518
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Knots having the same Alexander module which are not S-equivalent
As is discussed in this answer, S-equivalence class can be regarded as the Alexander module plus Blachfield pairing, an isomorphism of some duals of Alexander modules.
There are examples of knots ...
- 2,797
24
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1
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862
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The congruence subgroup property for mapping class groups and a conjecture of Grothendieck
This question is about a link between an open question in low-dimensional topology and a conjecture of Grothendieck, proved by Mochizuki. Let's start by stating them.
Recall that a subgroup $K$ of a ...
- 25k
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3
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On a metrized $n$-dimensional manifold $X$, does every $x \in X$ have a small ball $B_\delta(x)$ that is homeomorphic to $\mathbb R^n$?
Suppose that $X$ is an $n$-dimensional topological manifold that is also metrizable, and hence equipped with some metric that induces the topology.
For every point $x \in X$, let $B_\delta(x)$ be the ...
- 5,327
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1
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The importance of Poincare Conjecture or SPC4?
As well known, Perelman proved Poincare conjecture by proving Thurston's Geometrization conjecture.
Somebody says that we can understand part of the universe from Poincare conjecture.
As a purely ...
- 5,230
8
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1
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349
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Finite two-relator groups and quotients of knot groups
Let $G$ be a one-relator group $\langle A \mid R = 1 \rangle$. Then clearly $G$ is finite if and only if it is cyclic of finite order, i.e. can be given by a presentation $\langle a \mid a^n = 1 \...
- 25k
10
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443
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Analytic continuation gives a covering space (and not just a local homeomorphism)
Let $\mathcal{G}$ be the space of germs of holomorphic functions defined on open subsets of $\mathbb{C}$, topologized in the usual way. There is a natural map $p\colon \mathcal{G} \rightarrow \mathbb{...
- 148k
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1
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445
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What is known/expected on the co-growth series of the braid group?
The co-growth series of finitely generated group with respect to generating set $S$ is generating function for the number of words of length $n$ which are equal to 1 in the group.
Its studies ...
- 1,819
1
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1
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216
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Partitions of unity with arbitrary Lip-constants
Lets make things simple. Suppose we have a compact metric space $(X,d)$ and then some Lipschitz partition of unity exists, say a collection $\mathcal{F}=\{f_n\}$ subordinate to some open cover $\...
CommunityBot
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9
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1
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401
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Cohomological gap in arithmetic groups
$\DeclareMathOperator\SL{SL}$For the sake of this question, let's say that a group $G$ of finite cohomological dimension $n$ has a cohomological gap if for some $0 < i < n$ there is no subgroup $...
- 12.3k
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120
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Can every orientation preserving homeomorphism of a manifold isotoped to be identity on a locally flat embedded disk?
Let $M$ be a connected (topological) oriented $m$-manifold (say without boundary), and
let $\operatorname{Homeo}^+(M)$ be the group of orientation preserving homeomorphisms $M \to M$.
Is it true that ...
- 1,726
3
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0
answers
93
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Finiteness of non-orientable 3-manifolds with the same orientable two-fold cover
Given a compact, orientable and boundary incompressible 3-manifold $M$. Suppose that either $M$ is closed, or $\partial M$ consists of tori.
For which non-orientable 3-manifolds $N$, the orientable ...
- 605
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0
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65
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Approximating curves using only line sections and arcs
Given a curve $C$, I need to construct another curve $C'$ approximating $C$ under the following constraints: (1) $C'$ needs to be smooth, (2) $C'$ is composed of only line sections and arcs, (3) $C'$ ...
CommunityBot
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