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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1answer
119 views

On Thurston's triangulations of sphere

I have two questions from Thurston's paper [1]. In the paper [1], Thurston talks about classifying certain classes of triangulations of the sphere. Here a triangulation of a sphere a Topological ...
5
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1answer
134 views

Homotopy in $X$ and homology in $X \times I$

Suppose $X^n$ and $M^{n-2}$ are manifolds, and $f_1,f_2 : M \to X$ to two homotopic embeddings of $M$ into $X$. We can then embed $M$ into both boundary components in $X \times I$ using $f_1$ and $...
3
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1answer
107 views

Geometry of a manifold after Dehn filling, in terms of geometry pre-filling

First time posting, so sorry if this is an uninteresting or overly long post! The inspiration for this question was sparked by this answer given by Bruno Martelli in response to a question about ...
1
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2answers
90 views

Model for random graphs where clique number remains bounded

In the Erdös-Rényi model for random graphs,the clique number is seen to go to infinity al the number of vertices grows. Is anyone aware of models for random graphs with bounded ...
4
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1answer
173 views

Cobordism modelling fibration over $S^1$

Let $X$ be a closed oriented manifold which is a fibration over $S^1$ whose fiber $F$ is connected, i.e. $X\cong F\times[0,1]/\sim h$, for an $h\in \mathrm{Diff}(F)$. Suppose that $b_1(X)=1$. ...
10
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0answers
149 views

Vietoris-Rips complex and coarse geometry

Let $K$ be an infinite countable subset of Euclidean space $E$ such any point of $E$ is within distance 1 of some point of $K$. In the language of John Roe's "coarse geometry", this implies that $K$ ...
2
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1answer
98 views

Easy lemma for trivalent graphs in colored Jones polynomial

In his 2008 paper, Tanaka, Toshifumi, The colored Jones polynomials of doubles of knots, J. Knot Theory Ramifications 17, No. 8, 925-937 (2008). ZBL1149.57023. Tanaka stated a lemma (Lemma 3.3) ...
6
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0answers
58 views

Second cohomology of handlebody mapping class group

Let $H$ be a genus $g$ handlebody, let $\mathrm{Mod}(H)$ be its mapping class group. Is the calculation of $H^2(\mathrm{Mod}(H),\mathbb{Z})$ known? (Let $S$ be the boundary of $H$, then $\mathrm{Mod}(...
3
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1answer
91 views

Are Seifert fibered spaces with a horizontal surface exactly the surface bundles over the circle with periodic monodromy?

Are Seifert fibered spaces with a horizontal surface exactly the surface bundles over a circle with periodic monodromy? I am unsure of my arguments for this: If a SFS has a horizontal surface then ...
6
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1answer
118 views

Starting letters of equivalent infinite geodesic paths of hyperbolic Coxeter groups

Let $\left(W\text{, }S\right)$ be a Gromov hyperbolic Coxeter system and denote by $\partial W$ the corresponding Gromov boundary. For $z\in\partial W$ let $\alpha$, $\beta$ be infinite geodesic paths ...
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0answers
63 views

Cyclic homotopies of quotients of $S^3$

We are given a free action of an abelian finite group on $S^3$. Let $L$ denote the quotient space and let an element $\alpha \in \pi_1 L =G$ be given. Does there exist a cyclic homotopy $h_t:L \to L$ ...
12
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1answer
229 views

Is there a Morita cocycle for the mapping class group Mod(g,n) when n > 1?

Write Mod(g,n) for the mapping class group of a genus-$g$ surface $\Sigma$ with $n$ boundary components. When $n=0,1$ we define the Torelli group $T$ to be the subgroup of Mod(g,n) which acts ...
3
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0answers
99 views

Knot and its embedded disk

Let $K \subset S^3$ be an arbitrary knot. Let $D$ denote the embedded disk in $B^4$ bounded by $K$. Up to diffeomorphism, is it possible to describe the followings (at least for some trivial knots, ...
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1answer
424 views

Examples of hyperbolic groups

What are some other classes of word-hyperbolic groups other than the finite groups, fundamental groups of surfaces with Euler characteristics negative and virtually free groups?
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67 views

Branched 2-fold covering over edge of a 3-orbifold

I am reading the paper "On some generalized triangle groups and three-dimensional orbifolds" by Vinberg, Mennike and Khelling (Tran. Moscow Math. Soc. 1995 (56)). Let $k,l,m>0$, at most one of ...
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0answers
79 views

Trajectory definition using constrained projections on unknown surface

In $3D$ space where the $Z$ axis is up-down, I have the following: A static camera $A$ at $(x_a, y_a, z_a)$; A laser pointer $B$ at $(x_a, y_a, z_a + b)$ which can yaw or pitch by $1^\circ$ at a ...
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115 views

What is the connection between $\mathrm{AdS}_2$ and the hyperbolic plane $\mathbb{H}^2$?

What is the connection between $\mathrm{AdS}_2$ and the hyperbolic plane $\mathbb{H}^2$? Some sources seem to imply that they are the same, i.e. having at least the same symmetry group $\mathrm{SL}(2,...
6
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1answer
262 views

unlinking in 5 dimensions

If I have a linked pair of circles in $\mathbb{R}^3$, they can be unlinked in $\mathbb{R}^4$. Said differently, there is an isotopy in $\mathbb{R}^4$ between two strands which have been twisted, and ...
5
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0answers
94 views

Finitely generated nilpotent groups as cusp groups

I recently learned about the following question, asked by I. Kapovich : Is there an example of a group $G$ which is hyperbolic relative to some parabolic subgroups that are nilpotent of class $\geq 3$...
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1answer
167 views

Distinguishing Square Knot and Granny Knot using Quandles

It is known that the square knot and the granny knot are nonequivalent although they have isomorphic fundamental groups. I want to write a work on knot theory and provide these knots as an example ...
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0answers
68 views

Maximum number of ways of splitting a set of points with an hyperplane

Given a set $S$ of $n$ points in $\mathbb{R}^d$, let $D_S$ be the set $\{\mathbf{v}=|\mathbf{u}-\mathbf{u'}|: \mathbf{u},\mathbf{u'}\in S\}$ (where $\forall i=1,2,\ldots, d$, $\mathbf{v}_i=|\mathbf{u}...
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0answers
93 views

Quadrilateral fundamental domain

Let P be a hyperbolic quadrilateral. Poincare polygon theorem provides sufficient condition for P to be a fundamental domain of some Fuchsian group in term of its inner angles. I find the angle sum ...
7
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3answers
283 views

Wildness of codimension 1 submanifolds of euclidean space

This question arose out of this stack exchange post. I am wirting a thesis about the $s$-cobordism theorem and Siebenmann's work about end obstructions. Combined they give a quick proof of the ...
7
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1answer
275 views

To find a point in Teichmüller space or measured foliation, how many lengths of curves do you need?

To parametrize Teichmüller space, it suffices to measure the hyperbolic lengths of a finite number of curves. It is well-known that $9g-9$ curves suffice, by a standard pair-of-pants argument given in,...
14
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1answer
480 views

Simplicial set of permutations

Let $S_k$ be the set of all permutations of $k+1$ elements $0,1,...,k$. introduce boundary maps $d_i : S_k \rightarrow S_{k-1}$ by deleting from permutation $\eta$ element $\eta(i)$ and monotone ...
4
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2answers
191 views

Tiling of genus 2 surface by 8 pentagons

In theses these notes, Example 5.6, it is said that there is a "symmetric tiling of a genus 2 surface by 8 right-angled hyperbolic pentagons". Question 1: What does this tiling look like? Question 2:...
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0answers
165 views

Diffeomorphism classification of Grassmannian manifolds

Is anything known about the diffeomorphism classification of Grassmannian manifolds? I know that there are some results on projective spaces (for example in Lopez de Medrano's "Involutions on ...
2
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0answers
100 views

Is this a typo in Ihara's “On discrete subgroups of the two by two projective linear group over p-adic fields”?

In Eq. (9'') on p. 227 of Ihara's paper "On discrete subgroups of the two by two projective linear group over p-adic fields" (link), where the second line says $$"\log Z_{\Gamma}(0,\chi)=1",$$ is this ...
12
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3answers
437 views

Immersions of surfaces in $\mathbb{R}^3$

Stephen Smale famously proved in [Trans. Amer. Math. Soc. 90 (1959), 281-290] that any two $C^2$ immersions $S^2\to\mathbb R^3$ are regularly homotopic. This is how we knew that one can do a sphere ...
5
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196 views

Why does the longitude correspond to Frobenius in Arithmetic Topology, and other strange phenomena

I am trying to adress Morishita's book Knots and Primes to discover a bit about Arithmetic Topology, but some analogies puzzle me. I know that the correspondence should be addressed with a grain of ...
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1answer
47 views

Two multi-curves in a surface with the same transverse measure

Let $(\cal F,\mu)$ be the stable measured foliation of a pseudo-Anosov on an oriented surface $S$. Can there be two non-isotopic multi-loops (collections of disjoint simple loops) $L_1,L_2\subset S$, ...
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0answers
134 views

Category of Manifolds and Maps: TOP $\supseteq$ TRI $\supseteq$ PL $\supseteq$ DIFF? [closed]

Please let me denote the following (TOP) topological manifolds https://en.wikipedia.org/wiki/Topological_manifold (PDIFF), for piecewise differentiable https://en.wikipedia.org/wiki/PDIFF (PL) ...
7
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1answer
152 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 ...
5
votes
1answer
180 views

Curvature and asphericity of cube complexes

Let $K$ be a connected cube complex (one may assume that its a cellulation of a smooth, closed manifold). Such a $K$ comes equipped with a length metric (one assumes that each edge is of unit length). ...
6
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1answer
708 views

Cobordism Theory of Topological Manifolds

Unfortunately, due to my ignorance, my present knowledge is limited to the cobordism Theory of Differentiable Manifolds. Cobordism Theory for DIFF/Differentiable/smooth manifolds However, there are ...
9
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1answer
376 views

What do absolute neighborhood retracts look like?

In the course of filling in my map of non-pathological topology, I'd like to understand the class of ANRs (Absolute Neighborhood Retracts) as a sort of "neighborhood" of the class of CW complexes. ...
8
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0answers
107 views

Non-additivity of intersection forms

Given two oriented $4k$-manifolds $X_1$ and $X_2$, Novikov additivity tells us that $$ \sigma(X_1 \sharp X_2) = \sigma(X_1) + \sigma(X_2).$$ More generally, if we glue the boundaries of two such ...
6
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1answer
212 views

What are orbifolds with corners?

What is the geometric definition of orbifolds with corners? Here “geometric" means that there is a definition in chapter 8 of the draft of Dominic Joyce's book D-manifolds and d-orbifolds: a theory of ...
8
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0answers
142 views

Representation of the space of lattices in $\Bbb R^n$

The space of 2D lattices in $\Bbb R^2$ can be represented with the two Eisenstein series $G_4$ and $G_6$. Each lattice uniquely maps to a point in $\Bbb C^2$ using these two invariants, and the points ...
4
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2answers
189 views

The handlebody decomposition of S^1 bundles over surfaces?

What is the most natural handlebody decomposition of $F_g \times S^1$, if $F_g$ is an orientable closed surface of genus $g$?
6
votes
1answer
139 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 $...
11
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0answers
99 views

Which presentation of a 3-manifold group comes from a Heegaard splitting?

Given a balanced presentation of a three-manifold group G, is there any way to determine whether it comes from a Heegaard splitting? In other words, how to characterize such presentations? There exist ...
3
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0answers
98 views

What is topology of all Square matrices such that matrix times it transpose is diagonal

What is the topology of the subspace $X_n\subset M_n(\mathbb R)$ consisting of all non-zero $n\times n$ matrices $A$ such that $A^t A$ is diagonal ? For example $X_2$ is the product of a torus and an ...
6
votes
1answer
180 views

Can an exotic diffeomorphism of the 4-ball change the isotopy class of an embedded surface?

Let $W$ be $B^4$ or $S^3 \times I$. Let $Y$ be a properly embedded surface in $W$. Let $f : W \to W$ be a diffeomorphism which is the identity near $\partial W$. Very little is known about $\pi_0(\...
5
votes
1answer
109 views

Complete folds and one cut

The fold-and-cut theorem states that any shape with straight sides can be cut by a single complete straight cut if the paper is the folded flat in the right way. Here is an example from an answer on ...
3
votes
1answer
175 views

Wall self-intersection invariant for odd-dimensional manifolds?

I am trying to convince myself that a naïve definition of the Wall self intersection number should not work for odd-dimensional manifolds. Namely, let $X^{2n-1}$ be a smooth oriented closed manifold ...
11
votes
1answer
239 views

Do (3+1)-dimensional Lorentzian manifolds admit unique smoothings?

Of course, 3-dimensional topological manifolds admit unique smoothings while 4-dimensional topological manifolds generally do not. A (3+1)-dimensional topological Lorentzian manifold (definition below)...
3
votes
1answer
156 views

Obtaining the bounding 4-manifold from the Heegaard diagram

It is well known that any orientable closed 3-manifold $M$ admits an Heegaard splitting $M = H_1\cup H_2$ where $H_i$ is an handlebody of genus $g$. It is also well known that such an $M$ is the ...
7
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0answers
149 views

Disciplining dunce hats

I'm wondering if anyone has a copy of a preprint by Charles Giffen from 1977, with the enjoyable title, Disciplining dunce hats in 4-manifolds. I've seen it referred to in various places, including ...
3
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0answers
481 views

Pointwise stabilizer of an apartment of the Bruhat-Tits building of $\mathrm{SL}_n(\mathbb{Q}_p)$

Denote by $X$ the Bruhat-Tits building of $\mathrm{SL}_n(\mathbb{Q}_p)$. Let $\Sigma$ be the fundamental apartment of $X$. Let $\Gamma=\mathrm{SL}_n(\mathbb{Z}[\frac{1}{p}])$. We can prove that the ...