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).

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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 ...
Igor Rivin's user avatar
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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 ...
Michael Freedman's user avatar
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 $CAT(0)$ ...
Luis Jorge's user avatar
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 ...
user1831's user avatar
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 ...
Rohil Prasad's user avatar
  • 1,591
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 ...
Cusp's user avatar
  • 1,703
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 ...
YCor's user avatar
  • 60.1k
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 $...
cooper90's user avatar
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 ...
M. Dus's user avatar
  • 1,900
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 ...
user101010's user avatar
  • 5,319
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 ...
Piotr Hajlasz's user avatar
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 ...
Cezar's user avatar
  • 11
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 ...
user avatar
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 $A_{41}$, ...
Nicomachus's user avatar
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 ...
Piotr Hajlasz's user avatar
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 ...
giulio bullsaver's user avatar
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 ...
user101010's user avatar
  • 5,319
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 ...
Adam's user avatar
  • 2,370
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 ...
Junhyeong Kim's user avatar
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 ...
aglearner's user avatar
  • 14k
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 ...
Adam's user avatar
  • 2,370
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 ...
Adam's user avatar
  • 2,370
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 ...
Adam's user avatar
  • 2,370
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 ...
hakunamatata's user avatar
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 $\beta_k$ ...
Henry.L's user avatar
  • 7,951
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 (...
algebrachallenged's user avatar
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 ...
Yankes's user avatar
  • 147
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 $...
Angy's user avatar
  • 61
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 $X$. ...
Just Me's user avatar
  • 343
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. ...
QGravity's user avatar
  • 969
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 ...
John Wiltshire-Gordon's user avatar
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 ...
Thomas's user avatar
  • 2,691
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 $Y'$, ...
Adam Levine's user avatar
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 ...
G.Tverisovskikh's user avatar
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 $G$ ...
Janusz Przewocki's user avatar
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) \ ...
Ryan Budney's user avatar
  • 42.9k
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 $p$ ...
Nick L's user avatar
  • 6,923
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. ...
Michael Freedman's user avatar
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!
Markiff's user avatar
  • 303
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 ...
A. S.'s user avatar
  • 51
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 ...
I.Cast's user avatar
  • 33
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 ...
John's user avatar
  • 113
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: ...
user84144's user avatar
  • 2,769
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 ...
Nicolas Boerger's user avatar
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. ...
Christopher King's user avatar
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 ...
Stefan Friedl's user avatar
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 ...
user avatar
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'...
Christopher King's user avatar
3 votes
1 answer
415 views

Centralizer of a generator in a braid group

Given a braid group $$ B_n \simeq \left\langle x_1,\ldots,x_{n-1} \middle| \begin{array}{l} x_ix_j = x_jx_i, \;\text{for } |i-j|>1 \\ x_ix_{i+1}x_i = x_{i+1}x_ix_{i+1} \end{array} \right\rangle $$...
Anton Menshov's user avatar
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 ...
David G. Stork's user avatar

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