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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Complete geodesics on hyperbolic a pair of pants

I have asked this question on MSE. But I think Mo is a better place to ask my question. Here is the link to my question on MSE. I will rewrite it here: I am trying to understand the article by Maryam ...
Amirhossein's user avatar
7 votes
1 answer
364 views

Implications of Geometrization conjecture for fundamental group

Hempel proved that Haken manifolds have residually finite fundamental groups. With the Geometrization conjecture, this now holds for any compact and orientable 3-manifold. How exactly does the ...
aceituna's user avatar
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1 answer
444 views

Twisted spin bordism invariants in 5 dimensions

[Note]: My question will be a bit long. So, first, thank you for your careful reading, generous comments, helps and answers, in advance! The spin $G$-bordism invariant can be twisted in the way that ...
wonderich's user avatar
  • 10.3k
7 votes
2 answers
450 views

Is the *unreduced* Burau representation unitary?

In a 1984 paper, Squier wrote down an explicit Hermitian form $J$ relative to which the reduced Burau representation is unitary. The form is valued in the ring $\mathbb{Z}[s^{\pm 1}]$, where $s^2 = t$ ...
Nick Salter's user avatar
  • 2,790
7 votes
2 answers
567 views

What is the strongest known RSW result in planar percolation?

One of the weakest estimates conjectured to hold for critical planar percolation models (and proved in many cases) is the so-called RSW estimate. RSW estimate is the statement that the probability of ...
John Pardon's user avatar
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7 votes
1 answer
635 views

Fundamental group of the space of maps into a classifying space

Let $P : E \to X$ be a principal $G$-bundle, where $G$ is a connected topological group. $P$ is classified by a map $f: X \to BG$. The group of gauge transformations $\mathcal{G}$ of $P$ is defined to ...
Bilateral's user avatar
  • 3,064
7 votes
3 answers
312 views

Cyclic groups acting on balls, and interior fixed points

Let a finite cyclic group $G = \mathbb Z/n$ act continuously on an open $d$-ball $B^d$. Suppose further that this action extends to the closed ball $\overline{B^d}$. Is there necessarily a fixed point ...
Jens Reinhold's user avatar
7 votes
2 answers
285 views

Equivalence of definitions of quasiconformal surfaces?

I have been reading John H. Hubbard's book Teichmüller Theory vol. 1 and I am a little bit concerned with his definition of quasiconformal surface. Definition: A quasiconformal surface $S$ is a ...
Maxime Scott's user avatar
7 votes
1 answer
719 views

More on completion/compactification of open manifolds

This is a follow up to one of my previous questions (81714). Suppose that $M$ is an open manifold, say with a single end. Previously, I was concerned with realizing $M$ as the interior of a compact ...
Igor Khavkine's user avatar
7 votes
2 answers
691 views

Euler class of S^1-orbibundle

Given a topological space $M$ with a locally free $S^1$ action on it, assume the slice representation holds,(this is often the case, e.g. M is a smooth manifold) then this will make $M$ a principal $S^...
Shisen Luo's user avatar
7 votes
1 answer
243 views

Does the continuous image of a disc contain an embedded disc?

Let $\phi:\Bbb D^2\to\Bbb R^n$ be a continuous mapping of the 2-disc $\Bbb D^2$ that is injective on the boundary $\partial\Bbb D^2=\Bbb S^1$. Does its image contain an embedded disc with the same ...
M. Winter's user avatar
  • 12.5k
7 votes
2 answers
400 views

Examples of homology sphere that bound a nonsmoothable contractible 4-manifold

Freedman’s theorem shows that all 3-dimensional homology spheres bound topologically a contractible 4-manifold. It is well known that the Poincaré homology sphere does not bound a smooth contractible ...
J. GE's user avatar
  • 2,593
7 votes
1 answer
257 views

Stallings' binding tie

I came to know that the statement below could be proved using Stallings' binding tie argument, though I have no reference article proving the statement by the binding tie argument. Can anyone help me ...
Random's user avatar
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7 votes
2 answers
291 views

Groups acting on products of hyperbolic spaces

I am interested in groups acting properly and cocompactly by isometries on finite products of Gromov-hyperbolic metric spaces. I am mostly interested in the case where the group itself is not ...
Thomas Haettel's user avatar
7 votes
1 answer
215 views

Singularities of PL embedding of surface in a contractible 4-manifold

I am trying to understand the article "A solution to a conjecture of Zeeman" by Akbulut, but I am not an expert in PL-geometry. As far as I understand, two statements should be true, but I ...
P. Tolo's user avatar
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7 votes
2 answers
239 views

What are the "correct" references for the Vassiliev invariant?

Is there a good survey paper which describes the general ideas of Vassiliev's invariant? I am not an expert on knot theory, many references are too technical for me. Could Vassiliev's invariants be ...
user8749's user avatar
7 votes
1 answer
235 views

Diffeomorphisms pushing forward vector field to any non-zero multiple

Is there a closed smooth manifold $M$ such that for each real $x\neq 0$ there is a nowhere vanishing vector field $v$ on $M$ and a diffeomorphism $\phi:M\to M$ such that $\phi_*v=xv$?
user avatar
7 votes
1 answer
323 views

Fibers of continuous maps of $\mathbb{R}^n$ which are injective at dense points

Question. Suppose that $f\colon\mathbb{R}^n \to \mathbb{R}^n$ is a continuous map and there is a dense subset $D \subset \mathbb{R}^n$ such that $f^{-1}(f(x)) = \{x\}$ for all $x \in D$. Is every ...
Shinpei Baba's user avatar
7 votes
1 answer
631 views

Bers' constant for compact hyperbolic surfaces with geodesic boundary

The clasical Bers' theorem about pants decomposition says that any compact Riemann surface of genus $g \geq 2$ has a pants decomposition such that every cutting geodesic in this decomposition is of ...
Carabaev's user avatar
  • 295
7 votes
2 answers
587 views

Which topological spaces contain dense simply connected subspace?

And when can this subspace be chosen to be open? As the answer to this question indicates, any manifold contains an open dense subset, which is homeomorphic to $\mathbb{R}^{n}$, and so for manifolds ...
erz's user avatar
  • 5,385
7 votes
1 answer
299 views

Can a surface group act on a finite-valence simplicial tree?

Question. Let $S$ be a closed surface of genus $> 1$. Can $\pi_1(S)$ act faithfully and minimally on a simplicial tree of finite valence? Here "minimal" means that there is no invariant sub-tree. ...
Dylan Thurston's user avatar
7 votes
1 answer
176 views

Determine if an $n$-dimensional mesh of simplices is a non-manifold

In an $n$-dimensional space I have a set of simplices where each simplex consists of facets. Some of the simplices are 'connected' by sharing facets. Each facet is made up on edges, each consisting ...
Jörg's user avatar
  • 71
7 votes
1 answer
624 views

Characterisation of Q-rank 1

I'm looking for a reference and/or the original source for the following fact: An irreducible non-uniform lattice in a semisimple Lie group without compact factor has Q-rank 1 if and only if it does ...
ThiKu's user avatar
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7 votes
1 answer
457 views

Does every simplicial polytope have a topology-preserving contractible edge?

An edge of a triangulated manifold is said to be contractible if it may be contracted to a vertex without modifying the topological type of the underlying manifold. Otherwise, the edge is ...
Anand Kulkarni's user avatar
7 votes
1 answer
366 views

Exotic homeomorphisms of a cube

If $\varphi:\mathbb{R}\to\mathbb{R}$ is continuous, non-constant, non-decreasing, and differentiable a.e. with $\varphi'=0$ a.e., then the mapping $$ \Phi(x,y)=(x+\varphi(x),y+\varphi(y)) $$ is a ...
Piotr Hajlasz's user avatar
7 votes
1 answer
323 views

Does Freedman's disk embedding theorem extend to infinitely many immersed disks?

I'm referring to a statement of the disk embedding theorem on P17 in a book of Behrens-Kalmar-Kim-Powell-Ray, the Disk Embedding Theorem. See below. There are some new formulations, e.g., Theorem 1.2 ...
Shijie Gu's user avatar
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7 votes
1 answer
472 views

How to prove the product of Whitehead manifold and $\mathbb{R}$ is homeomorphic to $\mathbb{R}^4$?

I am currently reading Rolfsen's "Knots and Links". At page 82 Whitehead manifold $W$ is defined and an exercise asking to show that $W\times \mathbb{R}\cong \mathbb{R}^4$ is left. Reference ...
Math Diego's user avatar
7 votes
1 answer
278 views

Open covering of $S^n$ by sets not containing antipodal points

Given an $n$-dimensional sphere $S^n$ and an open cover such that none of the open sets contain antipodal points, does there exist a point on $S^n$ that belongs to at least $n+1$ open sets from the ...
Alan Li's user avatar
  • 71
7 votes
1 answer
317 views

Decomposition of manifolds with toroidal boundary

Let $\mathcal{M}$ be a compact, connected, oriented 3-manifolds with non-empty connected boundary $\partial\mathcal{M}$. Then, following this article, it is stated that $\mathcal{M}$ can be written as ...
G. Blaickner's user avatar
  • 1,137
7 votes
1 answer
353 views

Virtually large groups of small rank (related to 3-manifolds)

Edited 25.05.21: the assumptions of the question were incorrect, but as the discussion may be helpful for future MOnauts, I'll strike my mistakes and add clearly marked explanations afterwards. I am ...
lemon314's user avatar
  • 323
7 votes
1 answer
228 views

Retracting off a compact set

Let $K$ be a compact set in $\mathbb{R}^n$ and let $U$ be a bounded open set that contains $K$. You may assume both are connected. Can we always find an open $V$ such that $K\subset V\subset\...
erz's user avatar
  • 5,385
7 votes
1 answer
196 views

A coincidence between the Lambert cube, Lobell polyhedron, and hyperbolic 3-manifolds?

I. Lambert cube $\mathfrak L(\alpha_1,\alpha_2,\alpha_3)$ In this paper (p.8), we find the volume $V$ of the hyperbolic Lambert cube for the special case $\alpha=\alpha_1 = \alpha_2 = \alpha_3$ as $...
Tito Piezas III's user avatar
7 votes
1 answer
382 views

Pontryagin square and $\frac{1}{2}(\mathcal{P}(x) -x^2) =x \cup_1 Sq^1 x$

The Pontryagin square, maps $x \in H^2({B}^2\mathbb{Z}_2,\mathbb{Z}_2)$ to $ \mathcal{P}(x) \in H^4({B}^2\mathbb{Z}_2,\mathbb{Z}_4)$. Precisely, $$ \mathcal{P}(x)= x \cup x+ x \cup_1 2 Sq^1 x. $$ ...
annie marie cœur's user avatar
7 votes
1 answer
361 views

Lickorish-Wallace theorem for torsion spin$^c$ 3-manifolds?

The Lickorish-Wallace theorem tells us that any closed 3-manifold $Y$ is an integer link surgery on $S^3$, which yields an oriented cobordism between $S^3$ and $Y$. Filling out the $S^3$ by a 4-ball $...
cjackal's user avatar
  • 355
7 votes
1 answer
272 views

boundary of semihyperbolic groups

There are various definitions of boundary of a hyperbolic group. Which of those generalize to semi-hyperbolic groups (in the sense of Alonso and Bridson)? The example I have in mind is a semisimple ...
Roman's user avatar
  • 1,506
7 votes
1 answer
303 views

Can a 3-ball divide a standard 4-ball into two exotic 4-balls?

Let $B^n$ denote the unit ball in $\mathbb{R}^n$ (wrt the standard euclidean metric) and $\bar{B}^n$ denote the unit closed ball. Suppose that $\Sigma$ is a a smooth embedded hypersurface with ...
foliations's user avatar
  • 1,119
7 votes
1 answer
220 views

Why is the dividing set nonempty when a convex surface has Legendrian boundary?

I am an undergrad and curious about the following question. Let $(Y,\xi)$ be a contact manifold, and $L\subset (Y,\xi)$ be a Legendrian knot which is the boundary of a convex surface $\Sigma$ embedded ...
contakto's user avatar
7 votes
1 answer
527 views

Standard (special) spines and hyperbolic structure on 3-manifolds

My question relates to constructing angled triangulations or hyperbolic triangulations for $3$--manifolds. Briefly, an angle triangulation can be considered as an assignment of a real number (called ...
Don Shanil's user avatar
7 votes
1 answer
785 views

Negative intersection of symplectic submanifolds

For a symplectic 4-manifold, it is possible for two symplectic submanifolds to intersect negatively? Actually it is an exercise in the 4-manifold book by Gompf and Stipsicz to find symplectic planes ...
Hwang's user avatar
  • 1,388
7 votes
2 answers
487 views

Computations of the Link homology categorifying the second colored Jones polynomial

Has anybody done computations of such a theory? Is there a place I could look up and see what the answers are for low crossing knots?
Charlie Frohman's user avatar
7 votes
1 answer
2k views

Ambiguous definition of "nerve of an open covering" on wikipedia?

Let $(U_i)_{i\in I}$ be an open covering of a topological space $X$. At http://en.wikipedia.org/wiki/Nerve_of_an_open_covering, the nerve of the open covering is defined as follows: the nerve $N$ ...
Rasmus's user avatar
  • 3,144
7 votes
1 answer
354 views

Two details from Stallings's proof of the sphere theorem

EDIT: After a little prompting by Mark Grant, I answered the first question in the comments. The second question remains open. Let $M$ be a compact $3$-manifold with $\pi_2(M) \neq 0$. The sphere ...
Laura's user avatar
  • 343
7 votes
1 answer
337 views

Decidability of knot equivalence in general 3-manifolds? Surface equivalence?

Given a closed orientable 3-manifold $M^3$ and two knots $K_1$ and $K_2$ in $M$, is there an algorithm to decide if $K_1$ and $K_2$ are isotopic? Is there an algorithm to decide if there is a ...
user101010's user avatar
  • 5,319
7 votes
1 answer
207 views

Can the Chern-Simons invariant of a cusped hyperbolic $3$-manfiold be defined mod $\mathbb Z$?

For a closed hyperbolic $3$-manifold $M$, the Chern-Simons invariant $CS(M)$ can be defined as an element of $\mathbb R/\mathbb Z$. When $M$ is cusped it can still be defined, but is now only defined ...
Calvin McPhail-Snyder's user avatar
7 votes
1 answer
297 views

Assigning a "canonical geometry" to a Seifert surface

I originally posted this on stackexchange, but it hasn't gotten an answer. I hope it's not inappropriate for this forum. Suppose I have a knot $K: S^1 \hookrightarrow S^3$ with minimal genus Seifert ...
gdd's user avatar
  • 175
7 votes
1 answer
359 views

Thickness and hierarchical hyperbolicity

Thick metric spaces were introduced by Behrstock, Drutu and Mosher, see here. Hierarchically hyperbolic spaces were introduced by Behrstock, Hagen and Sisto, see here. I've heard that it is open ...
M. Dus's user avatar
  • 1,900
7 votes
1 answer
554 views

Can two small exotic smooth $\mathbb{R}^4$ manifolds be combined as a standard smooth $\mathbb{R}^4$?

I just seen De Michelis and Freedman's paper Uncountably many exotic $\mathbf{R}^4$'s in standard 4-space, J. Differential Geometry 35 (1992) pp 219-254, doi:10.4310/jdg/1214447810. If I understand ...
Shou-Jyun Zou's user avatar
7 votes
1 answer
172 views

Minimum number of double points over all immersed disks

Let $K$ be a knot in the boundary of a compact smooth 4-manifold $X$, and suppose that $K$ is the the kernel of $\pi_1(\partial X) \to \pi_1(X)$. Then $K$ is the boundary of some immersed disk $D \to ...
user101010's user avatar
  • 5,319
7 votes
1 answer
270 views

Are isotopic and conjugate homeomorphisms, conjugate by an element in $\mathrm{Homeo}_0(M)$?

An answer to this question would also answer Isotopy of periodic homeomorphisms of a surface along periodic homeomorphisms Let $M$ be a topological manifold and let $f,g$ be two orientation ...
Paul's user avatar
  • 1,379
7 votes
1 answer
652 views

What is "topology in dimension 3.5"?

I've noticed a couple of conference titles which reference something called "topology in dimension 3.5," such as this one and this one. This subject seems quite mysterious to me — it looks like ...
Arun Debray's user avatar
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