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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Reference request and prerequisites for understanding the Sphere Theorem and the Loop Theorem in 3-manifold theory

As part of my directed studies project, my advisor has suggested that I completely understand the proof of the Sphere Theorem and the Loop Theorem in 3-manifold theory and explain it to him. I have ...
ZSMJ's user avatar
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2 votes
2 answers
185 views

Fibration of hyperbolic 3-manifold

A fibration of a manifold $\phi: M \to S^1$ gives rise to a short exact sequence $$ 1 \to \pi_1(N) \to \pi_1(M) = \mathbb{Z} \overset{f_\ast}{\to} 1 $$ where $N$ is the fiber. I've heard that, if $M$ ...
return true's user avatar
2 votes
1 answer
325 views

What are the best definitions for smoothness of a 2D curve (real-valued function)?

Sounds like a trivial question, but could not find any answer other than the fact that there are many ways to define it. My problem is this: I look at different elevation maps, some with sharp ...
Vincent Granville's user avatar
1 vote
0 answers
156 views

Membership test of convex set

Let $K$ be a compact convex subset of $R^n$ which has some positive gaussian measure, say at least 1/2. For each nonzero vector $u \in R^n$, we define another compact convex set $K * u$ in the ...
Sandra's user avatar
  • 11
9 votes
1 answer
363 views

Morse theory on outer space via the lengths of finitely many conjugacy classes

Let $F_n$ be the free group on letters $\{x_1,\ldots,x_n\}$ and let $X_n$ be the (reduced) outer space of rank $n$. Points of $X_n$ thus correspond to pairs $(G,\mu)$, where $G$ is a finite connected ...
Sarah's user avatar
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2 votes
1 answer
251 views

Entanglement, quadrics and $\mathbb{P}^2(\mathbb{C}^3)$ [closed]

First of all: I apologise in advance for if my question will be arid, wrong written or even nonsensical. I was at a talking with a professor last week, and the question of "Entanglement and ...
Red Bordeaux's user avatar
2 votes
0 answers
428 views

What are some of the big open problems in $4$-manifold theory?

I've recently been studying some Manifold Theory and got very interested in their topological as well as geometric properties. From my understanding of the current literature, most the big and ...
sadman-ncc's user avatar
9 votes
1 answer
336 views

Mapping class groups are finitely generated

Let $N$ be a compact smooth manifold. By "mapping class group" I will mean $$\pi_0 \operatorname{Diff}(N)$$ i.e. the isotopy-classes of diffeomorphisms of $N$. My presumption is that this ...
Ryan Budney's user avatar
  • 42.8k
6 votes
0 answers
191 views

Is Heegaard-Floer homology the Lagrangian Floer homology of $M=\text{Sym}^g(\Sigma), L_0=\mathbb{T}_{\alpha},L_1=\mathbb{T}_{\beta}$?

Is Heegaard Floer homology the Lagrangian Floer homology of $M=\text{Sym}^g(\Sigma), L_0=\mathbb{T}_{\alpha},L_1=\mathbb{T}_{\beta}$? I am interested in the relationship between the theories ...
contingent's user avatar
17 votes
0 answers
993 views

"Next steps" after TQFT?

(Disclaimer: I'm rather nervous that this isn't appropriate for MathOverflow, but given the contents of my question I don't really know a better place to ask something like this.) Recently, I've been ...
Nicholas James's user avatar
2 votes
1 answer
139 views

English version of a paper by Gusarov

I am looking for the english translation of the paper in russian Variations of knotted graphs, geometric technique of n-equivalence, St. Petersburg Math. J. 12-4 (2001) by Gusarov. There is a .ps file ...
bd99's user avatar
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4 votes
1 answer
188 views

Conformal map between flat and hyperbolic torus with a boundary

I am confused because I can define two very different complex structures on the torus with a puncture/boundary. For my first construction, I can imagine removing a disk from a flat torus, inheriting ...
Holomaniac's user avatar
1 vote
1 answer
216 views

A torus bundle whose vertical tangent bundle is indecomposable

I would like to see an example of a fiber bundle $\mathbb{T}^2 \to X \xrightarrow{\pi} B$ whose fibers are tori $\mathbb{T}^2:=(S^1)^2$ and whose vertical tangent bundle $\ker(d\pi) \to X$ is not ...
Anon's user avatar
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3 votes
0 answers
112 views

Integrating over a fundamental domain in $\text{SL}(d,\mathbb R)$ under $\text{SL}(d,\mathbb Z)$ action and the choice of the fundamental domain

Let $\mathcal{F}$ be a fundamental domain in $\text{SL}(d,\mathbb R)$ under $\text{SL}(d,\mathbb Z)$ action. It is well known that there exists a unique $\text{SL}(d,\mathbb R)$-invariant probability ...
user506835's user avatar
4 votes
0 answers
326 views

Does a contractible locally connected continuum have an fixed point property?

I'm surprised that I can't find any research on this topic. Maybe it's too obvious? Kinoshita proved that contractible continuum do not have FPP, but his example is not locally connected. Maybe if we ...
LoliDeveloper's user avatar
3 votes
1 answer
290 views

A detail in Brown's proof of the generalized Schoenflies theorem

Consider a homeomorphic embedding $h:S^{n-1}\times [0,1]\rightarrow S^n$ and denote $$S^{n-1}_t=h(S^{n-1}\times \{t\}).$$ The generalized Schoenflies theorem states the closure of each connected ...
Nikhil Sahoo's user avatar
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8 votes
1 answer
205 views

Integral homology classes of which no multiples admit embedded representatives with trivial normal bundle

Let $M$ be a closed smooth manifold of dimension $n$ and $z\in H_l(M,\mathbb{Z})$ a $k$-dimensional integral homology class. Theorem II.4 of Thom's classical 1954 paper states that for $l< n/2$ or $...
Zhenhua Liu's user avatar
6 votes
1 answer
216 views

How small need a perturbation be to not change the diffeomorphism type of a variety?

Let $f,g \in \mathbb{R}[x_0,\dots,x_k]$ be homogeneous polynomials and $X:=Z(f) \subset \mathbb{RP}^k$ be the projective variety defined by $f$. Assume that $X$ is smooth and has codimension $1$. Then ...
user505117's user avatar
1 vote
1 answer
159 views

Identifying a curve on a closed surface of genus 4

The notation is the one used in the attached picture. Take a closed, orientable surface $\Sigma_4$ of genus $4$, obtained as the identification space of a polygon with $16$ sides in the usual way. The ...
Francesco Polizzi's user avatar
4 votes
0 answers
54 views

Kernel of the geometric intersection form

Let $\Sigma$ be a closed surface and $\mathcal C$ the set of all free homotopy classes of closed (may be nonsimple) curves in $\Sigma$. Consider the geometric intersection form $i$ on $\mathbb Z\...
nim's user avatar
  • 357
0 votes
1 answer
162 views

Mappings of reducible 3 manifolds with boundary

In section 3 of his paper "Mappings of reducible 3 manifolds" McCullough, proves that every self-homeomorphism of a reducible 3 manifold can up to isotopy be written as a composition of ...
ThorbenK's user avatar
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5 votes
1 answer
226 views

Cancellation of elements in the Gromov boundary of a free group

Let $A$ be a finite set of free generators and their inverses and $F$ the free group generated by elements in $A$ (some call $A$ the alphabet of $F$). For each $g\in F$, use $\vert\,g\,\vert$ to ...
Sanae Kochiya's user avatar
3 votes
0 answers
287 views

"Maehara-style" proof of Jordan-Schoenflies theorem?

The highest upvoted answer to this old question Nice proof of the Jordan curve theorem? is a proof by Ryuji Maehara. I personally really liked/appreciated that Maehara's proof is A) a fairly ...
D.R.'s user avatar
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2 votes
0 answers
134 views

Characterization of the homotopy type of the $C^\infty$ topology

One of the subtle aspects of geometric topology is the interaction of function space topologies and homotopy theory. There are many reasonable topologies to put on the space of smooth embeddings $\...
Connor Malin's user avatar
  • 5,191
10 votes
2 answers
273 views

Reference request - Fibrations between spaces of embeddings

This is a cross-post of this question from MSE. Given topological manifolds $M$ and $N$ of the same dimension, let $\operatorname{Emb}(M,N)$ denote the subspace of $\operatorname{Map}(M,N)$ ...
Ken's user avatar
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4 votes
1 answer
392 views

4-manifold $M$ with intersection form of Leech lattice

Do we know any 4-manifold $M$ with intersection form of rank-24 Leech lattice? Is $M$ smooth? Differentiable to which $n$-order? Can $M$ be obtained by any surgery on 3 copies of E8 4-manifolds with ...
zeta's user avatar
  • 337
5 votes
1 answer
253 views

0-surgery on a fibered hyperbolic ribbon knot

Does there exist a fibering hyperbolic ribbon knot such that the 0 surgery is exceptional? If so does there exist such an example where the result of 0-surgery is Seifert fibered? I tried looking at ...
ThorbenK's user avatar
  • 1,175
6 votes
1 answer
318 views

Does $\overline{\mathbb{C}P^2 \setminus B^4}$ embed (smoothly/topologically) in $\mathbb R^5$?

Question: Does $\overline{\mathbb{CP}^2 \setminus B^4}$ (that is the closure of complex projective plane minus a 4-ball) embed (smoothly/topologically/piecewise linearly) in $\mathbb R^5$? Background: ...
Martin Tancer's user avatar
2 votes
1 answer
125 views

Space of the trivial long knot in the thickened surface

Let $F$ be a compact oriented surface and $x_0\in F$ a basepoint. Consider the set $\mathcal E=Emb_0(I,F\times I)$ of embeddings $\sigma\colon I\hookrightarrow F\times I$, $\sigma(\partial I)=\{x_0\}\...
nim's user avatar
  • 357
6 votes
0 answers
132 views

S¹ action on a manifold which generates "non-torsion" loop in diffeomorphism group

I am interested in $S^1$-actions on smooth, closed, and oriented manifolds $M$. I suppose that the action has a fixed point (I also suppose $M$ is connected). Let $\operatorname{Diff}(M)$ denote the ...
onefishtwofish's user avatar
5 votes
3 answers
204 views

First usage of the terms pseudo-isotopy and concordance in manifold theory

I am hoping I can use the collective knowledge of the forum to piece together some history. I'm wondering where the terms pseudo-isotopy and concordance originated, in their modern forms as used in ...
Ryan Budney's user avatar
  • 42.8k
0 votes
2 answers
302 views

Reference for a topological result

I am reading the short paper due to Erdös and Bollobás "On a Ramsey-Turán type problem", where they obtain a lower bound for the number of edges on an $n$-graph without $K_4$ as a subgraph ...
Johnny Cage's user avatar
  • 1,543
7 votes
2 answers
633 views

A generic metric on $X\cup\mathbb Z$

$\newcommand\abs[1]{\lvert#1\rvert}$Let $(X,d_X)$ be a countable metric space such that $X\cap\mathbb Z=\{0\}$. Problem. Is there a metric $d$ on the union $Y=X\cup\mathbb Z$ such that $d(x,y)=d_X(x,...
Taras Banakh's user avatar
  • 40.8k
3 votes
2 answers
446 views

Finding a hyperbolic metric with geodesic boundary on a given Riemann surface

Let $X$ be a Riemann surface with analytic boundary. Assume that $X$ has negative Euler characteristic. Then there exists a conformal hyperbolic metric $X$ such that $\partial X$ consists of geodesics ...
Yuxiao Xie's user avatar
6 votes
1 answer
395 views

Abstract simplicial complexes - Reference for an elementary definition of mapping degree for simplicial maps?

I am interested to use the mapping degree for simplicial maps between (oriented) abstract simplicial complexes. What I mean by "elementary": My preference would be to use a definition of ...
Claus's user avatar
  • 6,777
4 votes
2 answers
335 views

Knot theory in handlebodies of arbitrary genus

It is well known that not all graphs embed on the plane (e.g. the graph $K_{3,3}$). However, every finite graph embeds into a surface of some genus. One can think of this procedure as starting with a ...
João Lobo Fernandes's user avatar
2 votes
0 answers
229 views

A Question about an article by Birman, Series

Birman and Series in their article GEODESICS WITH BOUNDED INTERSECTION NUMBER ON SURFACES ARE SPARSELY DISTRIBUTED proved that the set of points on a hyperbolic surface (possibly with boundary) ...
Amirhossein's user avatar
4 votes
1 answer
208 views

Euler class of vertical tangent bundle of the surface bundle over circle

Suppose $\Sigma$ is an oriented genus $g>1$ surface and $h:\Sigma\to \Sigma$ is a diffeomorphism preserving a point $p$. Let $M$ be the surface bundle over $S^1$ obtained by gluing $\Sigma\times I$ ...
Faniel's user avatar
  • 623
8 votes
1 answer
420 views

dichotomy in hyperbolic groups

Suppose $G$ is a word hyperbolic group i.e. every geodesic triangle in a cayley graph with respect to a finite generating set of $G$ is $\delta$-thin, for some $\delta>0$. There are various ...
ggt001's user avatar
  • 141
1 vote
1 answer
165 views

Bring's curve $\sum_{i=1}^5 x_i^k = 0$ for $k = 1,2,3$ and an analogue $\sum_{i=1}^6 y_i^k = 0$ for $k = 1,2,4,7$

Bring's curve or Bring's surface with genus 4 and $5!=120$ automorphisms can be given by the homogeneous equations, $$x_1+x_2+x_3+x_4+x_5 = x_1^2+x_2^2+x_3^2+x_4^2+x_5^2 = \\x_1^3+x_2^3+x_3^3+x_4^3+...
Tito Piezas III's user avatar
2 votes
0 answers
141 views

Can distinct meridians commute in a knot group?

Suppose I have a knot $K$ in $S^3$. Given a diagram $D$ of $K$ I get the Wirtinger presentation $\langle x_1, \dots, x_a \mid r_1, \dots, r_c\rangle$ of its knot group $\pi(K) = \pi_1(S^3 \setminus K)$...
Calvin McPhail-Snyder's user avatar
1 vote
1 answer
95 views

PL embedding of 3-manifold PL triangulation into 5-manifold PL triangulation

Are there any PL triangulations of closed orientable 3-manifolds that do not PL-embed into some PL triangulation of a closed 5-manifold?
user49988's user avatar
  • 111
3 votes
0 answers
178 views

What's the meaning of this relation between volumes of $n$-balls and umbral calculus?

The volume of an $n$-ball of radius $1$ is $$V_{n}={\frac {\pi ^{n/2}}{\Gamma {\bigl (}{\tfrac {n}{2}}+1{\bigr )}}}.$$ The functional equation of Riemann zeta function is $${\displaystyle \pi ^{-{s \...
Anixx's user avatar
  • 9,306
6 votes
1 answer
463 views

A characterization of metric spaces, isometric to subspaces of Euclidean spaces

I am looking for the reference to the following (surely known) characterization of metric spaces that embed into $\mathbb R^n$: Theorem. Let $n$ be positive integer number. A metric space $X$ is ...
Taras Banakh's user avatar
  • 40.8k
4 votes
0 answers
151 views

Examples of Lattices of Sp(n,1)

$Sp(n,1)$ is the isometry group of $n$-dimensional quaternionic hyperbolic space. It is written in literature that the group is an example of a hyperbolic groups. Can you suggest me any reference ...
ggt001's user avatar
  • 141
2 votes
0 answers
94 views

Lifting homology classes to the unit tangent bundle, a la Johnson

Let $M$ be a oriented smooth closed 2-manifold, and let $\gamma$ be an oriented smooth simple closed curve in $M$. In Spin structures and quadratic forms on surfaces, Johnson definines a standard way ...
Tanny Sieben's user avatar
8 votes
1 answer
303 views

The growth rate of a commutator set in a non-elementary group

Let $G$ be a non-elementary group generated by a finite set $S$. Here, a group is called non-elementary if it is not virtually abelian. Denote $S^{\le n}:=\{g\in G: |g|_S\le n\}$ for any $n\in \mathbb ...
dennis's user avatar
  • 145
3 votes
1 answer
139 views

Cohomology of cocompact lattices in hyperbolic spaces

I have a (maybe too naive) hope that cocompact torsion-free arithmetic lattices in hyperbolic spaces $X \neq \mathbb{H}_\mathbb{R}^2$ are uniquely determined by their cohomology with coefficients in $\...
TSU's user avatar
  • 131
10 votes
3 answers
610 views

Doubles of 2-handlebodies

Let $X$ denote a $4$-manifold with boundary obtained by adding $k_1$ $1$-handles to $B^4$ and $k_2$ many $2$-handles to the resulting manifold i.e. $X$ is an arbitrary $4$-dimensional $2$-handlebody. ...
ThorbenK's user avatar
  • 1,175
1 vote
1 answer
148 views

Tiling the hyperbolic plane by non-regular quadrilaterals

We add a bit to Which polygons tessellate the hyperbolic plane?. Question: Are there hyperbolic quadrilaterals with all angles different (not necessarily irrational fractions of π) that tile the ...
Nandakumar R's user avatar
  • 5,473

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