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).
4,134
questions
6
votes
1
answer
288
views
Generating prime knots (in order)
In this really cool paper https://arxiv.org/abs/1612.03368, A. Malyutin shows that the probability that a random prime knot of up to $N$ crossings (as $N$ goes to infinity) is not generically ...
8
votes
0
answers
197
views
Slicing satellite knots
Call a knot L a "braid-pattern satellite" of a knot K if it is a satellite of K and the pattern on which it is based is a closed braid in the solid torus. Is there a knot K so that no braid-pattern ...
6
votes
1
answer
326
views
Bruhat-Tits building of $SL_n(\mathbb{Q})$, hyperbolic isometries and its axis
Consider $G=SL_n(\mathbb{Q})$ and $p$ a prime integer. Associated to $G$ and $p$ we have its Bruhat-Tits building $\Delta$.
It is well known that $\Delta$ can be provided with a canonical $CAT(0)$ ...
3
votes
1
answer
139
views
Is there a geometric interpretation of a Zariski dense surface subgroup?
Is there a geometric interpretation of a surface subgroup being Zariski dense? Or, conversely, given a $\Pi_1$ injective surface in a 3-manifold, is there a geometric or topological requirement on the ...
39
votes
4
answers
4k
views
Thurston's "tinker toy" problem
In the article "On Being Thurstonized" by Benson Farb (located here), a particular result of Thurston is mentioned.
Namely, suppose a "tinker toy" $T$ is a contraption consisting of a multitude of ...
4
votes
2
answers
309
views
Nielsen-Thurston decomposition from the product of Dehn twists
Given a closed surface of genus $g\geq 2$, we know that the mapping class group $Mod(S)$ is generated by the Dehn twists. My question is
Given an element as a product of Dehn twist, is it possible ...
17
votes
1
answer
360
views
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 ...
2
votes
1
answer
126
views
Teichmuller uniqueness theorem with marked points
Let $S$ be a genus $g$, $g > 1$ Riemann surface, and let $h \colon S \to S$ be a homeomorphism of $S$. We denote by $[h] \in \text{Map}(S)$ the corresponding element of the mapping class group of $...
10
votes
1
answer
684
views
Parabolic subgroups of relatively hyperbolic and CAT(0) groups
Let $G$ be a finitely generated group. We say that $G$ is CAT(0) if it acts properly and co-compactly by isometries on a CAT(0) space.
We say it is hyperbolic relative to a collection $\Omega$ of ...
3
votes
1
answer
141
views
An algorithm to tell if two cut systems are handle slide equivalent?
Let $\Sigma$ be a genus $g$ closed orientable surface and let $\alpha = \{ \alpha_1,...,\alpha_g\}$ and $\beta = \{ \beta_1 ,..., \beta_g \}$ be two cut systems (i.e. nonintersecting homologically ...
10
votes
0
answers
403
views
Lipschitz homotopy groups
There is an extensive literature on Lipschitz homotopies of Lipschitz maps. But I haven't seen anything about Lipschitz homotopy groups. We have introduced this notion in an article that you can find
...
1
vote
1
answer
81
views
Equation for a geometrical half surface from folding a flat curved in one direction surface in half [closed]
I cannot not formulate the problem as i do not know how to model this idea. This is an open question not an mathematical exercise so if anybody has a good proposition i'm happy to use it.
Sorry in ...
2
votes
0
answers
49
views
Finite translation surfaces with Veech groups that are non-elementary Fuchsian groups of the second kind?
I know that all Veech surfaces have Veech groups which are Fuchsian groups of the first kind and that there exist finite translation surfaces with Veech groups that are elementary Fuchsian groups of ...
2
votes
0
answers
96
views
Union of Two Faces, using the Jordan Curve Theorem
Consider four disjoint points in the plane, $v_{1}$, $v_{2}$, $v_{3}$ and $v_{4}$.
The cycle, $C:=v_1v_2v_{3}v_{4}v_{1}$, is the union of the (Jordan)
arcs, $A_{12}$, $A_{23}$, $A_{34}$, and $A_{41}$, ...
14
votes
3
answers
1k
views
Linking topological spheres
Is there a simple proof of the fact that:
If $A\subset S^3$ is homeomorphic to $S^1$, then there is a circle $B$
embedded into $S^3\setminus A$ that such that the circles $A$ and $B$
are ...
1
vote
0
answers
60
views
Annuli and pinched annuli vs circles and horocycles
Any annulus is biholomorphich to the Poincare' disk $D$ from wich a circle centered at the origin has been removed.
If we want to parametrise annuli with punctures at one boundary, give the punctures ...
8
votes
1
answer
283
views
Obstructions to realizing a balanced presentation as a 3-manifold group
I am thinking of 3-manifolds as arising from Heegaard splittings which I am thinking about in terms of Heegaard diagrams. I know that 3-manifold groups are rather special in the class of all finitely ...
5
votes
1
answer
238
views
Can every curve be made transversal to a foliation by applying a pseudo-Anosov?
Let $F$ be a compact oriented surface with a foliation $\cal F$ with $k$-prong singularities only (or, if it helps, assume that $\cal F$ admits an invariant measure). Is it true then there exists a ...
2
votes
1
answer
131
views
Construction of self-covering map of any surface
Let $\Sigma(g,n)$ be an $n$-punctured surface of genus $g$.
If we assume that $f:\Sigma(g,n)\rightarrow\Sigma(g,n)$ is a branched self-covering map of degree $d$, then the equality follows from the ...
5
votes
1
answer
142
views
Short basis in $\pi_1$ on a hyperbolic surface of bounded diameter
First, some terminology. Let $(S,x)$ be a compact surface of genus $g>0$. A standard collection of loops $\gamma_1,\ldots, \gamma_{2g}$ based at $x$ is a collection of loops that cuts $S$ into a ...
1
vote
0
answers
108
views
Powers of pseudo-Anosov and the geometric intersection numbers
Let $\phi$ be a pseudo-Anosov of a compact oriented surface $F$ with boundary. Let $\beta\subset F$ be a simple closed loop and $\alpha$ either a simple closed loop or an embedded arc with endpoints ...
2
votes
1
answer
192
views
Putting a transverse measure on a surface foliation
Let $F$ be an orientable surface with a foliation $\cal F$ with $k$-prong singularities only, for $k\geq 3$.
Since I am looking for an invariant transverse measure on $\cal F$, assume that there is ...
3
votes
1
answer
119
views
Are isotopic transversal curves on a foliated surface transversally isotopic?
Let $F$ be an orientable surface (possibly with boundary) with a foliation $\cal F$ with $k$-prong saddle singularities only for $k\geq 3,$ (as in figure borrowed from Farb-Margalit book). Suppose ...
1
vote
0
answers
122
views
Flows commuting with Anosov flows and further reference request
Hello respected members of Mathoverflow. I was reading the paper "Flots d’Anosov dont les feuilletages stables sont différentiables" by Etienne Ghys and there was a statement which he remarked was ...
7
votes
0
answers
258
views
Relations between Betti numbers for clique complex
Given a clique complex $K$ constructed from a discrete set of vertices (i.e. its faces are isomorphic to the set of cliques in the 1-skeleton of $K$.), it seems that the Betti numbers $\beta_k$ ...
6
votes
2
answers
384
views
General position for map from surface to 3-manifold
Let f be a smooth map from a (compact,oriented) surface S to a (compact, oriented) 3-manifold M. Suppose that I have an embedded (non-contractible) loop $\gamma$ in my surface $S$, can I find an (...
2
votes
1
answer
446
views
Square Peg Problem counterexample
Inscribed square problem:
Every continuous simple closed curve in the plane
contains four points that are the vertices of a
square.
I thinking about possibility of creating counter example to ...
2
votes
1
answer
251
views
Confusion about Teichmuller curves and $SL_2$ action
Let $M_g$ be the moduli space of curves, $\Omega M_g$ the total space of the bundle of holomorphic 1-forms and $\pi: \Omega M_g\to M_g$ the natural projection. On $\Omega M_g$ there's an action of $...
9
votes
0
answers
220
views
Fixed-points of a topological circle action
Suppose the circle group $G = S^1$ acts on $X$.
If $X$ is a closed smooth manifold (and the action is smooth), then we know the fixed-points $X^G$ are a disjoint union of smooth submanifolds of $X$. ...
7
votes
2
answers
1k
views
Topological Classification of Four-Manifolds
It is known that the topological classification of a closed Riemann surface is determined by its genus. Similar statements are proven for other compact Riemann surfaces with boundaries/marked points. ...
38
votes
3
answers
2k
views
If $X$ and $Y$ are homotopy equivalent, then are $X \times \mathbb{R}^{\infty}$ and $Y \times \mathbb{R}^{\infty}$ homeomorphic?
Let $X$ and $Y$ be reasonable spaces. Since $\mathbb{R}^{\infty}$ is contractible,
$$
X \times \mathbb{R}^{\infty} \cong Y \times \mathbb{R}^{\infty} \;\;\; \implies \;\;\; X \simeq Y.
$$
Is the ...
15
votes
1
answer
596
views
What is this quotient of the triangle 2-3-7 group?
I have been working with Hurwitz groups, and I came across the group $G := \langle a, b \ | \ a^2, b^3, (ab)^7, ([a,b]^2ab)^6 \rangle$. I'm trying to figure out exactly what this group is. I know it ...
12
votes
0
answers
244
views
Simply connected homology cobordisms
I'm looking for interesting examples of a homology 3-sphere $Y$ for which there exists a smooth, simply connected homology cobordism from $Y$ to itself (or simply to another homology 3-sphere $Y'$, ...
2
votes
2
answers
151
views
How to prove that $\phi: \;\mathrm Mod(S_g)\to \mathrm Sp(2g, \mathbb{Z})$ is an epimorphism? [duplicate]
How do I prove that homomorphism $\phi : \; \mathrm{Mod}(S_g)\to \mathrm{Sp}(2g, \mathbb{Z})$ (induced by the action of mapping class group of a surface on integer homologies of a surface) is an ...
11
votes
1
answer
159
views
A group of type F that is an extension of type F-by-type F
Let us first recall that a group of type $F$ is a group admitting a compact classifying space.
Let $K$ and $Q$ be groups of type $F$. Consider the family $\mathcal{G}(K, Q)$ consisting of groups $G$ ...
8
votes
0
answers
173
views
Stratification of space of labelled circles in the plane
Consider the space of $n$ round circles in the plane to be the open subset of $\mathbb R^{3n}$:
$$C_n = \{ (v_1, v_2, \cdots, v_n, r_1, r_2, \cdots, r_n ) : v_i \in \mathbb R^2, r_i \in (0, \infty) \ ...
4
votes
1
answer
137
views
Can we perturb a surface away from an orbifold point?
Let $X$ be a smooth, compact, orbifold of dimension $4$, where the stabilisers are only allowed to be cyclic groups. Let $p \in X$ be an isolated orbifold point (i.e. the orbifold chart about $p$ ...
37
votes
3
answers
989
views
How to specify a compact topological 4-manifold with a finite amount of data
Finite data for 4-manifolds: I have a somewhat hazy memory of seeing a Ph.D. thesis around 2005 which showed that any compact topological 4-manifold $M$ can be specified by a finite amount of data. ...
5
votes
1
answer
378
views
closed geodesics are dense in both hyperbolic surface and unit tangent bundle of hyperbolic surface
Could you please recommend me some references for proofs of this fact: "closed geodesics are dense in both hyperbolic surface and unit tangent bundle of hyperbolic surface". Thanks in advance!
3
votes
0
answers
146
views
Diffeomorphisms preserving "nice" smooth functions
Let $\mathbb{R}^2\supset D=\{(x,y)\in\mathbb{R}^2|x^2+y^2<1\}$ be the open unit disc, and $U\subset\mathbb{R}^2$ be the interior of Koch's snowflake, as constructed in Falconer's book Fractal ...
3
votes
1
answer
240
views
Examples of (non-discrete) hyperbolic totally disconnected locally compact groups whose boundaries are spheres
I'am wondering whether there exists a non-discrete hyperbolic totally disconnected locally compact group such that the boundary is a finite-dimensional sphere. If the answer is positive, could you ...
1
vote
1
answer
144
views
Order question about pentagonal tiling type 9 and type 10
People found there were only existing 15 types of pentagonal tiling after one hundred years' work, see Pentagonal tiling.
These 15 types of pentagonal was named by finding date except type 9 and type ...
5
votes
2
answers
332
views
Criterion for alternation of the linking form
I was recently informed by a source of the following fact:
Theorem 1: The linking form on an orientable smooth 5-manifold $M$ is alternating if and only if $M$ is spin$^{\mathbb{C}}$.
Question 1: ...
10
votes
1
answer
984
views
Acyclic Finite Groups
A group is called acyclic if its classifying space has the same homology of a point. Examples of acyclic groups include Higman's group with four generators and relations, also ...
5
votes
2
answers
397
views
Smallest tile to *isohedrally* tessellate the hyperbolic plane
Is there a smallest tile (in terms of diameter) that isohedrally tessellates the hyperbolic plane?
In this question, we ask the same question without the isohedral requirement, and the answer was no. ...
20
votes
1
answer
1k
views
Topological embeddings of real projective space in euclidean space
I was wondering whether the real projective space $\Bbb{R}P^n$ embeds topologically into $\Bbb{R}^{n+1}$ for odd $n$.
It certainly doesn't for even $n$ because of Alexander duality. Also it doesn't ...
3
votes
0
answers
70
views
Does the orbital function divided by the volume of a ball decrease?
Let $X$ be a Cartan-Hadamard manifold, meaning a complete, connected, simply connected Riemannian manifold with non-positive sectional curvature and $\Gamma < Isom(X)$ a discrete group of ...
20
votes
2
answers
2k
views
Smallest tile to tessellate the hyperbolic plane
Is it known what the smallest tile (in terms of area) that can tessellate the hyperbolic plane is? In particular, it should tessellate the plane by itself.
I think it will be a Triangle group, but I'...
3
votes
1
answer
415
views
Centralizer of a generator in a braid group
Given a braid group
$$
B_n \simeq
\left\langle
x_1,\ldots,x_{n-1}
\middle|
\begin{array}{l}
x_ix_j = x_jx_i, \;\text{for } |i-j|>1 \\
x_ix_{i+1}x_i = x_{i+1}x_ix_{i+1}
\end{array}
\right\rangle
$$...
3
votes
2
answers
372
views
Minimum required crossings in a link diagram for a $k$-component Brunnian link
What is the minimum number, $s$, of crossings in a link diagram for $k$ (component) links fully knotted together such that cutting any single link frees all individual component links--becomes an ...