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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An incomplete characterisation of the Euclidean line?

We say that a metric space $(X, d)$ is a Banakh space if for every $\rho \in \mathbb{R}_{> 0}$ and every $x \in X$, there are $a,b \in X$ such that $\{y \in X \, \vert \, d(x, y) = \rho\} = \{a, b\}...
Luc Guyot's user avatar
  • 7,353
6 votes
2 answers
602 views

Powers of meridians in knot groups

Given a (tame) knot $K \subset S^3$, let $t \in G = \pi_1(S^3 - K)$ be any meridian. The Wirtinger presentation shows that $\langle \langle t \rangle \rangle = G$, where the notation indicates the ...
Joe Boninger's user avatar
2 votes
0 answers
114 views

Foliation of $X$ by once punctured planes without any singularities

Let $n=3.$ Take $X=(0,1)^n.$ Fix points $p,q$ s.t. $\text{dist}_n(p,q)=\sqrt{n}.$ Construct a smooth regular foliation of $X$ with $(n-1)-$dim. leaves which are topologically $(0,\sqrt{n})\times S^{n-...
53Demonslayer's user avatar
8 votes
2 answers
330 views

Is every contractible homogeneous space of a connected Lie group homeomorphic to a Euclidean space?

Problem. Let $G$ be a connected Lie group and $H$ is a closed subgroup of $G$ such that the homogeneous space $G/H$ is contractible. Is $G/H$ homeomorphic to a Euclidean space $\mathbb R^n$ for some $...
Taras Banakh's user avatar
  • 40.7k
4 votes
0 answers
180 views

Symmetric line spaces are homeomorphic to Euclidean spaces

For points $x,y,z$ of a metric space $(X,d)$ we write $\mathbf Mxyz$ and say that $y$ is a midpoint between $x$ and $z$ if $d(x,z)=d(x,y)+d(y,z)$ and $d(x,y)=d(y,z)$. Definition: A metric space $(X,d)$...
Taras Banakh's user avatar
  • 40.7k
3 votes
0 answers
93 views

Using blow-up for handle attachment

In basic surgery theory, it is clear what happens topologically when one "adds a handle", but getting the smooth structure right requires care. For example, in identifying opposite sides of ...
Dev Sinha's user avatar
  • 4,950
3 votes
0 answers
131 views

Isotopy-classes of oriented Schoenflies spheres in $S^4$ is a group under oriented connect-sum

Given two oriented, smoothly-embedded copies of $S^3$ in $S^4$ (called Schoenflies spheres), one can take an oriented connect-sum of the pairs $(S^4, M_1) \# (S^4, M_2)$. This puts a monoidal ...
Ryan Budney's user avatar
  • 42.8k
13 votes
1 answer
431 views

Compact closed aspherical manifolds with vanishing second homology in all the covering spaces

I wonder if there exists a compact closed smooth aspherical manifold $M$ of dimension at least $4$, so that for any covering space $\tilde{M}$ over $M,$ we always have $H_2(\tilde{M},\mathbb{Z})=0$ ...
Zhenhua Liu's user avatar
4 votes
2 answers
422 views

Finite application of one of Reidemeister moves on a knot diagram

It is known that given a knot diagram we can transform it into a trivial unknot diagram by a series of Reidemeister moves. The key word is "series". Can we transform any knot diagram using a ...
I. S.'s user avatar
  • 41
4 votes
1 answer
136 views

Topological type of complement of Heegaard curves in Heegaard surface $(\Sigma - \alpha - \beta)$

Suppose $(\Sigma, \alpha, \beta)$ is a genus-$g$ Heegaard diagram for a closed, oriented $3$-manifold $Y$, i.e. $\Sigma$ is an orientable genus-$g$ surface, and $(\alpha_1, \dots, \alpha_g)$ and $(\...
Matija Sreckovic's user avatar
7 votes
1 answer
221 views

Hyperbolic homology spheres with infinite $\mathrm{SL}_2(\mathbb{C})$ character variety

$\DeclareMathOperator\SL{SL}$By the celebrated results of Culler and Shalen, a closed $3$-manifold contains an incompressible surface if its $\SL_2(\mathbb{C})$ character variety is infinite. Now, for ...
Renaud Detcherry's user avatar
47 votes
3 answers
3k views

A metric characterization of the real line

Is the following metric characterization of the real line true (and known)? A nonempty complete metric space $(X,d)$ is isometric to the real line if and only if for every $c\in X$ and positive real ...
Taras Banakh's user avatar
  • 40.7k
14 votes
2 answers
444 views

Identify this 16 crossing knot

For this knot with DT code -8 16 28 14 -2 6 -20 10 24 -32 -12 -30 18 -22 4 -26 I received an error message from Knot Finder: "KnotFinder encountered an unknown error." I also tried SnapPy (...
Christoph Lamm's user avatar
2 votes
1 answer
215 views

How to get a presentation of the mapping class group of the $n$-punctured sphere

$\DeclareMathOperator\Mod{Mod}$I would like to compute the mapping class group (homeomorphism preserving orientation modulo those isotopic to the identity) of the sphere $S^2$ minus $n$ points $p_1,\...
Federico Fallucca's user avatar
7 votes
1 answer
268 views

Dupin cyclide as the stereographic projection of a Hopf torus

Let $H \colon S^3 \to S^2$ be the Hopf map and let $\gamma$ be a curve on $S^2$. Then $H^{-1}(\gamma)$ is called the Hopf cylinder or the Hopf torus when $\gamma$ is closed, with profile curve $\gamma$...
Stéphane Laurent's user avatar
5 votes
3 answers
530 views

If a polyhedron in $\mathbb{R}^3$ has local intersections, does it also have more global intersections?

Consider a simplicial complex $K$. A piecewise linear map $f: K \to \mathbb{R}^n$ is an almost-embedding if $f(\sigma) \cap f(\tau) = \emptyset$ for any two disjoint simplices $\sigma,\tau$ in $K$. ...
Omega Tree's user avatar
4 votes
1 answer
119 views

Inheritance of arithmeticity properties in orbifold strata

Suppose $M = K\backslash G/\Gamma$ is a quotient of a symmetric space by a lattice. I don't know all of the proper adjectives to apply here (e.g. what should be said about $G$ and so on), but I wouldn'...
Ethan Dlugie's user avatar
  • 1,247
4 votes
0 answers
162 views

Every closed surface divides a closed 3-manifold $X$ into two parts if and only if $H_1(X)$ is finite

My friend is looking for proof of the following statement Every closed surface divides a closed 3-manifold $X$ into two parts if and only if $H_1(X; \mathbb{Z})$ is finite. Rumor source: Justin ...
Arshak Aivazian's user avatar
3 votes
0 answers
70 views

non-negative curvature condition for polyhedral manifolds

A polyhedral manifold P, i.e, a topological manifold with a triangulation where each simplex is isometric to a simplex in Euclidean space (other constant curvature spaces are allowed), is said to have ...
Lucas L.'s user avatar
3 votes
0 answers
99 views

Explicit parameterizations of complicated unlinks?

I have a somewhat empirical question which I hope is still welcome here. I would like to know how to write down explicit parameterizations of "complicated unlinks", say with 2 or 10 ...
Sprotte's user avatar
  • 1,045
14 votes
1 answer
565 views

How do we know there are no more Deligne–Mostow/Thurston lattices?

In the context of hypergeometric functions, Deligne and Mostow enumerated several lattices in complex hyperbolic space/the rank 1 Lie group $\operatorname{PU}(1,n)$ (see [1] and [2]). Thurston used ...
Ethan Dlugie's user avatar
  • 1,247
2 votes
0 answers
67 views

Maximal orders and surface subgroups of even genus

Let $A$ be a quaternion algebra over a totally real number field $k$. Suppose that $A$ splits at exactly one real place of $A$. Let $\mathcal{O}$ be a maximal order in $A$. Then $\mathcal{O}$ contains ...
Jacques's user avatar
  • 563
5 votes
1 answer
223 views

Amenable link groups

The unknot and the Hopf link are (as far as I know) the only links whose complements have abelian fundamental groups. Are there more examples whose complement have amenable fundamental group?
ThorbenK's user avatar
  • 1,175
2 votes
1 answer
184 views

What is known about the almost complex structure on the Teichmüller space in Fenchel–Nielsen coordinates?

There has been a question on the same subject, but I'm asking about something more specific. In the Fenchel–Nielsen coordinates, the Teichmüller space of genus $g$ is represented as $\mathbb{R}^{3g-3}\...
Yuxiao Xie's user avatar
5 votes
2 answers
335 views

Does every triangulable manifold have a vertex-transitive triangulation?

Does every triangulable manifold have a vertex-transitive triangulation? When I talk about a vertex-transitive triangulation of a manifold, I mean in the sense of realizing a manifold homeomorphically ...
Mike's user avatar
  • 335
2 votes
1 answer
191 views

Corollary in Rasmussen's paper about $s$-grading of Lee's canonical generators

In Jacob Rasmussen's paper Khovanov homology and the slice genus, he states as Corollary 3.6 that $s(\mathfrak s_o)=s(\mathfrak s_{\bar o})=s_{min}(K)$, where $s$ is the $s$-grading and $\mathfrak s_o,...
boink's user avatar
  • 213
4 votes
1 answer
145 views

Existence of tubular neighborhood of singular complex subvariety

Let $X$ be a smooth complex projective manifold and let $Y$ be a closed subvariety of $X$ with $y\in Y$ a fixed point. Does there exist an open neighborhood $U$ of $Y$ such that $\pi_1(Y,y)\to \pi_1(U,...
Higgs-Boson's user avatar
3 votes
1 answer
206 views

Handle attachment information from Morse function and triangulation

First, allow me to setup the relevant information. It is well known that a Morse function $f:M\to\mathbb{R}$ induces a handle decomposition of $M$. For simplicity, let's restrict for now to the ...
rab's user avatar
  • 139
7 votes
1 answer
175 views

Calculating the Seifert framing for an exceptional fiber in a Seifert-fibered integer homology 3-sphere

Let $Y=\Sigma(\alpha_{1},\dots,\alpha_{n})$ be a Seifert fibered homology 3-sphere, let $\pi:Y\to\Sigma$ denote the projection to the orbifold surface, and let $T\subset Y$ be the pre-image under $\pi$...
Ian Montague's user avatar
4 votes
1 answer
222 views

Bounds for the crossing number in terms of the braid index?

Is there a lower bound on the crossing number of a knot (resp., link) with braid index $b$? For knots, I believe the smallest crossing number for braid index 2 is 3, the smallest crossing number for ...
Charles's user avatar
  • 8,974
3 votes
1 answer
186 views

Maps of surfaces to CAT(0) cube complexes

Let $C$ be a locally CAT(0) cube complex. Let $f\colon \Sigma_g \rightarrow C$ be a continuous map from a closed oriented genus $g \geq 1$ surface to $C$. Is it always possible to find a cube ...
Ursula's user avatar
  • 31
3 votes
1 answer
306 views

Simplified Bing's house

Bing's house is an example of contractible 2-complex in $\mathbb{R}^3$. One may think that it is a surface without boundary that has two types of singularities: tripod curves — curves where three ...
Anton Petrunin's user avatar
10 votes
0 answers
197 views

Bi-Lipschitz mappings

Assume that we have a bi-Lipschitz mapping $f:\bar{\mathbb{B}}^n(0,1)\to\mathbb{R}^n$. The mapping need not be smooth anywhere and it may happen that it cannot be extended to a homeomorphism of a ...
Piotr Hajlasz's user avatar
3 votes
1 answer
345 views

Boundaries of subsets of simply-connected domains

I am trying to prove the following assertion: Let $B$ be a simply-connected set, and let $B' \subsetneq B$ be a proper open connected subset. Then, there exists a point $x \in \partial B'$ of the ...
travis schedler's user avatar
3 votes
0 answers
67 views

Discreteness of volumes of boundary-parabolic representations

Suppose $M$ is a cusped hyperbolic $3$-manifold of finite volume. Let $\mathfrak{R}_0(M)$ be the space of boundary-parabolic representations $\rho : \pi_1(M) \to \operatorname{PSL}_2(\mathbb C)$. Is ...
Calvin McPhail-Snyder's user avatar
11 votes
0 answers
224 views

The second coefficient of the Conway polynomial from Knot Floer homology

Let $\nabla_K(z)$ be the Conway polynomial and $\Delta_K(t)$ be the Alexander polynomial normalized by $\Delta_K(t)=\Delta_K(t^{-1})$ and $\Delta_K(1)=1$, These invariants are equivalent and they are ...
Tetsuya Ito's user avatar
7 votes
2 answers
254 views

Surjections from genus $n$ surface group to free group of rank $n$

Let $\Sigma_n$ be a genus $n$ surface, let $\mathcal{H}_n$ be a genus $n$ handle body, and let $F_n$ be a free group of rank $n$. Fix an identification of $\pi_1(\mathcal{H}_n)$ with $F_n$. I know ...
Annie's user avatar
  • 73
12 votes
1 answer
577 views

Definition of Thurston's skinning map

A key construction in Thurston's proof of the existence of hyperbolic structures on Haken manifolds is the so-called "skinning map" associated to a 3-manifold $M$ with boundary whose ...
mrburch's user avatar
  • 155
6 votes
1 answer
297 views

"Neck cutting" and why gauge theory doesn't work on homotopy 4-spheres

I attended a talk recently and the speaker said, essentially, that gauge theory invariants are expected to never be able to detect exotic 4-spheres because they always vanish, for a reason related to ...
Audrey Rosevear's user avatar
6 votes
2 answers
165 views

Generate $\mathrm{Mod}(S_g)$ by two Dehn twists

Let $S_g$ be a closed orientable surface of genus $g>1$. How can one prove that its mapping class group $\mathrm{Mod}(S_g)$ is not generated by two Dehn twists? A pair of simple closed curves in $...
Andrey Ryabichev's user avatar
0 votes
0 answers
86 views

Terminology for discrete subgroups of PSL(2,k), where k is a non-archimedean local field

$\DeclareMathOperator\PSL{PSL}$I'm asking about terminology for discrete subgroups of $\PSL(2,k)$, where $k$ is a non-archimedean local field. As it is rather clumsy to have to use such expressions ...
Hercule Poirot's user avatar
1 vote
0 answers
106 views

Instantons on the 4-sphere with respect to other Riemannian metrics

It is known that the moduli space of $\text{SU}(2)$ instantons of charge $1$ on $S^4$ is diffeomorphic to the five-ball $B^5$ if $S^4$ is endowed with the round metric. Question: what does the moduli ...
Shaoyun Bai's user avatar
5 votes
0 answers
119 views

Can every homeomorphism of the 4-ball fixing the boundary be approximated by diffeomorphisms smoothly isotopic to the identity?

Let $B^{n}$ denote the euclidean closed ball of dimension $n$. Alexander's trick shows that every homeomorphism of $B^{n}$ fixing $\partial B^n$ is $C^{0}$-isotopic to the identity via a boundary ...
asymmetriad's user avatar
0 votes
0 answers
73 views

Is there any functoriality of Stallings' twists?

Suppose $L$ is an oriented link and $L'$ is a link obtained from the Stallings' twists. Are there any functoriality of the twist operation on the links, such as functoriality between (1) homology ...
jhbaik's user avatar
  • 1
3 votes
0 answers
106 views

Finite homology of a homogeneous space

Let $\Gamma$ be a cocompact lattice in $\operatorname{SL}(2,\mathbb R)$ and $X=\operatorname{SL}(2,\mathbb R)/\Gamma$ be the underlying homogeneous space. Can the homology group $H_1(X,\mathbb Z)$ be ...
William of Baskerville's user avatar
5 votes
0 answers
88 views

Irreducible factors of the A-polynomial

The A-polynomial $A_K$ of a knot $K$ describes the irreducible "non-abelian" components of the $SL(2)$-character variety of $S^3-K.$ Does anyone know a knot K for which $A_K$ has more ...
Adam's user avatar
  • 2,370
3 votes
0 answers
283 views

Fundamental group of blow-ups

Let $M$ be a simply-connected compact complex manifold of dimension three and $C$ is a smooth complex curve in $M$. Let $M'$ be the blow-up of $M$ along $C$. My question is: Is $M'$ also simply-...
Basics's user avatar
  • 1,821
3 votes
1 answer
175 views

Squier's conjecture on Burau at roots of unity

In Squier's short, yet influential, paper about the Burau representation, he made two conjectures that might have provided a proof for the faithfulness of the Burau representation (which we now know ...
Ethan Dlugie's user avatar
  • 1,247
3 votes
1 answer
210 views

Example of a non $\pi_1$-injective, degree one, self-map of a three-manifold

All manifolds will be assumed to be closed, oriented, and connected. Let $f\colon M\to M$ be a map of degree $\pm 1$. It is not hard to show that $\pi_1(f)$ is surjective. What is an example of a non ...
Random's user avatar
  • 885
4 votes
0 answers
259 views

Does this "join-like complex" of $K_5$ and $K_3$ embed in $\Bbb R^4$?

Consider the following 2-dimensional CW-complex: its 1-skeleton is $K_8$, which we write as an edge-disjoint union $K_5\cup K_{5,3}\cup K_3$. Then for any two edges $ab\in E(K_5)$ and $cd\in E(K_3)$ ...
M. Winter's user avatar
  • 12.5k

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