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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Identifying two definitions of orientation on a vector space

Let $V$ be an $n$-dimensional real vector space. Here are two definitions of an orientation on $V$: A generator of the $1$-dimensional vector space $\wedge^n V$, up to multiplication by positive ...
Jean's user avatar
  • 103
4 votes
1 answer
134 views

Dehn surgery on $RP^2 \times S^1$

A standard example of Dehn surgery is obtaining $S^3$ from $S^2 \times S^1$. Consider a unknot $L$ wrapping the non-trivial cycle $S^1$ in $S^2 \times S^1$. We drill out a tubular neighborhood $T_{L} $...
Topology_Dummy_Boy's user avatar
1 vote
0 answers
88 views

Mixing of geodesic flow and rate of mixing

I know the following result which is if $X:=G/\Gamma$ where $G$ be a Lie group and $\Gamma$ is a lattice in $G$. Then geodesic floe $F=(g_t)$ is exponentially mixing i.e., there exist $\gamma > 0$ ...
User1723's user avatar
  • 307
1 vote
1 answer
69 views

Simple convergence of convex compact set implies Hausdorff convergence

I am wondering about the following : In $\mathbb{R}^n$, suppose you are given compact convex bodies $\left\{ C_k : k \geq 1 \right\}$ and $C$, such that for every $x \in \mathbb{R}^n$ $$ \mathbb{1}_{...
Anthony's user avatar
  • 41
8 votes
4 answers
500 views

Residual finiteness of hyperbolic 3-manifold groups

So the consequence of the geometrization (according to 3-manifold group note) is that any finite-volumed hyperbolic 3-manifold is residually finite. So the question is: Q1. If $M$ is an infinite-...
one potato two potato's user avatar
0 votes
0 answers
163 views

Different notions of formality

Quoting from https://en.wikipedia.org/wiki/Rational_homotopy_theory a simply connected complex is called formal if its cohomology algebra is a model (in the sense of Sullivan) for the cochain complex. ...
ThiKu's user avatar
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4 votes
1 answer
374 views

Hyperbolic three-manifolds that fiber over the circle

Let $f$ be a pseudo-Anosov mapping class of a closed, connected, and oriented genus $g > 1$ surface. Let $M(f)$ be the corresponding hyperbolic three-dimensional mapping torus of $f$. Is the length ...
user524868's user avatar
1 vote
0 answers
85 views

Extend a circle action on $3$-manifolds

Let $M$ be an oriented closed $3$-manifold equipped with an effective smooth circle action. Can we have a classification of all such $M$ such that there exists a $4$-manifold $N$ with $\partial N=M$, ...
Zhiqiang's user avatar
  • 881
3 votes
0 answers
75 views

The boundary regularity of a Teichmüller domain

By a Teichmüller domain, I mean the Bers embedding of a Teichmüller space (of a compact oriented surface of finite type) in a complex space. It is known that the boundary of a Teichmüller domain is ...
Mahdi Teymuri Garakani's user avatar
2 votes
0 answers
372 views

$$ \left(\frac{\text{Man}^{\text{fr}}}{\text{Cobordism}},\coprod,\times \right)\simeq \left((\text{Fin}^{\simeq},\coprod)^{\text{gp}},\times\right)?$$ [closed]

If we combine a theorem of Pontryagin and the Barratt-Priddy-Quillen theorem we get that both sides of $$ \left(\frac{\mathrm{Man}^{\mathrm{fr}}}{\mathrm{Cobordism}},\coprod,\times \right)\simeq \left(...
Ola Sande's user avatar
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3 votes
1 answer
152 views

Principal circle bundles over punctured $3$-sphere

Let $M$ be $S^3$ with $k$ disjoint open balls $D^3$ removed. Can we classify all principal circle bundles over $M$ such that the total space is simply-connected?
Zhiqiang's user avatar
  • 881
0 votes
1 answer
46 views

Let K be a compact set in a surface, U component of S-K, K'=S-U. K has finitely many components. Does every component of K' contains a component of K?

Let $S$ be a compact connected surface. Let $K$ be a compact subset of $S$ and suppose that $K$ has a finite number of connected components. Let $U$ be a connected component of $S \setminus K$ and ...
Fernando Oliveira's user avatar
4 votes
1 answer
126 views

Heegaard splitting of figure-8 knot complement

It is well-known that the figure-8 knot complement in $S^3$ can be described as a circle fibration of a once-punctured torus. Is there also a description of the figure-8 knot as a Heegaard splitting ...
Oblonski's user avatar
2 votes
1 answer
89 views

For a closed Riemannian manifold $M$, must the set of points with non-unique closest points to a closed submanifold $S$ of $M$ be of 0 volume measure?

Let $(M,g)$ be a closed (compact without boundary) Riemannian manifold of finite dimension, with the volume measure $\mu:= \mu(E):=\int_{E}dvol_g \forall E \in \mathcal{B}(M),$ the Borel sigma algebra ...
Learning math's user avatar
2 votes
1 answer
156 views

Order of a loop around a cone point

Let $M$ be a connected orientable two-dimensional orbifold with only cone points as singular points. Assume that $M$ has genus $\geq 1$. Let $\alpha$ be a loop around an order $p$ cone point. Can we ...
RKS's user avatar
  • 533
6 votes
1 answer
108 views

Knotted concordances of slice links

Are there any examples of a link $L$ such that: $L$ is (strongly) slice, meaning that there exists a properly embedded collection $C$ of $n=|L|$ disjoint annuli in $S^3\times [0,1]$ such that $C\cap ...
Alessio Di Prisa's user avatar
2 votes
0 answers
123 views

Mutants or not?

Two 4-tangles (drawn unneccesarily complicated to show how they are related - both are 6-tangles capped off with the same cap): (alternate version with ends at the same point) If I could turn over ...
Hauke Reddmann's user avatar
6 votes
1 answer
283 views

Slice knots in 3-manifolds

Is there a nonslice knot $K\subset S^3$ that is slice in some closed oriented $3$-manifold $Y$? Here, when we say $K$ is slice in $Y$, it means that when regarded as a local knot in $Y\times\{1\}$, $K$...
Qiuyu Ren's user avatar
3 votes
1 answer
216 views

Rational 4-tangles vs rational knots

The closure of a rational $4$-tangle is a rational knot. But is the converse true? We could tangle up even the unknot to a hopeless mess before cutting it up, and we could cut it were it "hurts ...
Hauke Reddmann's user avatar
9 votes
0 answers
212 views

Existence of $1$-separated and $(1-\varepsilon)$-dense set in metric spaces

Is it know which metric spaces $M$ do have the following property: there is $\varepsilon>0$ and a maximal $1$-separated set which is $(1-\varepsilon)$-dense? In other words, when does at set $S\...
Christian's user avatar
  • 779
2 votes
1 answer
166 views

Example of pseudo $3$-manifold without any shape structure

I'm reading Andersen and Kashaev's A TQFT from quantum Teichmüller theory and the following condition in their definition of admissible oriented triangulated pseudo $3$-manifold confused me: ...
Shana's user avatar
  • 237
4 votes
0 answers
141 views

Subdivision via poset maps and pullback

In the following, all posets and complexes are assumed to be finite. For a poset $P$ denote by $|P|$ its geometric realization or nerve (i.e. forming the order complex and taking its geometric ...
KoopaTroopa's user avatar
1 vote
0 answers
51 views

Clique complex of expander graphs simply connected?

Given an expander graph family (an injective sequence of graphs with uniformly bounded vertex degree and a Cheeger constant/Laplacian spectral gap uniformly bounded away from zero). Can the ...
Florentin Münch's user avatar
4 votes
1 answer
109 views

Rigidity/flexibility of Sol-structures on closed 3-manifolds

This is a follow-up to the question Rigidity/flexibility of Nil-, Sol-, $\widetilde{\rm SL}_2$- structures on closed 3-manifolds From the answers/comments there and from an excellent survey by Bonahon ...
Roman's user avatar
  • 173
4 votes
1 answer
173 views

For which quadratic number field, the algebraic integers are cusps for some Coxeter group?

Let $H^2=\{(x,y)\mid y>0\}$ be the hyperbolic upper-half plane. Let $K=Q(\sqrt{d})$ be a quadratic number field, and $\mathcal{O}_K$ be the ring of algebraic integers in it. Let $\Gamma=\Delta(p,q,...
zemora's user avatar
  • 545
1 vote
0 answers
68 views

Representing geodesic compactifications of $S^1\times \Bbb R$ as analytic sections over base (analytic) foliations

Given a smooth nested set of "partial" foliations $\mathcal F_{\alpha}=\big\lbrace e^{\frac{\alpha}{\log x}}: \alpha \in (1/k,k), x\in(0,1),k\in [1,\infty) \big\rbrace$ of $X^2=(0,1)^2$ with ...
53Demonslayer's user avatar
2 votes
1 answer
88 views

Simple curves on hyperbolic tori

In the paper "Simple curves on hyperbolic tori" by McShane and Rivin, they show that if $T$ is a hyperbolic once punctured torus, one can define a norm on the homology $H_1(T,\mathbb{Z})$ by ...
stupid_question_bot's user avatar
2 votes
1 answer
155 views

Guts of 3-manifolds for sutured manifolds and pared manifolds

I found the notion "guts of three-manifolds" unclear to me. There exists "sutured guts" and "pared guts" in the literature, the well definedness of both are vague to me. ...
Fredy's user avatar
  • 492
7 votes
1 answer
176 views

Rigidity/flexibility of Nil-, Sol-, $\widetilde{\rm SL}_2$- structures on closed 3-manifolds

It is known that closed spherical and hyperbolic 3-manifolds are rigid. I.e., if two such manifolds are diffeomorphic, then they are isometric (moreover, I think, that every diffeomorphism is isotopic ...
Roman's user avatar
  • 173
0 votes
1 answer
184 views

Fixed points free automorphisms of Teichmüller spaces

Let $\mathcal{T}_{g,n}$ be the Teichmüller space of a compact oriented surface of genus $g$ with $n$ marked points. Assume that $N:=3g-3+n>0$. Viewing $\mathcal{T}_{g,n}$ as a bounded domain in $\...
Mahdi Teymuri Garakani's user avatar
1 vote
0 answers
81 views

Fibre product and submersion of PL-manifolds

Let us consider the fibre product $M\times_{f,g} M'$ of $M \xrightarrow{f} N \xleftarrow{g} M'$. If $M,M'$ and $N$ are smooth manifolds and $f$ is a submersion, then $M\times_{f,g} M'$ is again a ...
KoopaTroopa's user avatar
9 votes
1 answer
372 views

Is every automorphism of $\mathrm{Aut}^+(F_2)$ induced by conjugation inside $\mathrm{Aut}(F_2)$?

$\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Inn{Inn}\DeclareMathOperator\GL{GL}\DeclareMathOperator\SL{SL}$Let $F_2$ be a free group of rank 2. There is a surjection $\Aut(F_2)\rightarrow \GL(2,...
stupid_question_bot's user avatar
2 votes
1 answer
224 views

An interior cone condition for Teichmuller spaces

Let $\mathcal{T}_{g,m}$ be the Teichmuller space of a compact oriented surface of genus $g$ with $m$ marked points. Consider it as a bounded domain in a complex space $\mathbb{C}^N$. Let $\xi$ be a ...
Mahdi Teymuri Garakani's user avatar
7 votes
1 answer
240 views

Computing $\pi_1$ of the complement of a non-singular plane curve

The following is a well-known fact: Theorem. The fundamental group of the complement of a non-singular curve of degree $d$ in the complex projective plane is cyclic of order $d$. This was further ...
Marco Golla's user avatar
  • 10.4k
3 votes
1 answer
90 views

Geodesic laminations on the 4-punctured sphere

Let $S_{0,4}$ be the 4-punctured sphere equipped with a hyperbolic metric of finite volume (thus all punctures are cusps). Consider $\gamma$ to be a simple geodesic of $S_{0,4}$ (not necessarily ...
Nelson Schuback's user avatar
2 votes
1 answer
184 views

Does the group of compactly supported diffeomorphisms have the homotopy type of a CW complex?

It is known that the group of diffeomorphisms of a compact manifold with the natural $C^{\infty}$ topology has the homotopy type of a countable CW complex. See for instance this thread: Is the space ...
Yasha's user avatar
  • 469
0 votes
0 answers
22 views

An auxiliary problem while constructing the system of Jordan sets on a plane

Let $\mathfrak{S}$ be a system of rectangles in $R^2$ of the form $[a,b]\times [c,d]$ where $a,b,c, d \in R$, $a<b$, $c<d$. Let $\mathfrak{A}$ be a system of simple sets based on $\mathfrak{S}$. ...
Alexander's user avatar
5 votes
0 answers
141 views

What are the finite quotients of the braid group?

Are all known finite quotients of the braid group given by reducing the Burau or Lawrence-Krammer representations mod $p$ and evaluating at some element in $\mathbb{F}_p$? I recently saw a paper ...
Harry Reed's user avatar
1 vote
1 answer
209 views

Submersion of real projective space into Euclidean space

The famous Whitney immersion theorem states that any real projective space $\mathbb{RP}^n$ can be immersed into $\mathbb{R}^{2n}$. However, I haven't found information about the submersion ...
GHG's user avatar
  • 173
10 votes
2 answers
554 views

Prove these are not surface groups

For $g,n \geq 1$, let $\Gamma_{g,n}$ be the group with the following presentation: $$\langle \text{$a_1,b_1,\ldots,a_g,b_g$ $|$ $[a_1,b_1]^n [a_2,b_2] \cdots [a_g,b_g]=1$} \rangle.$$ For $n = 1$, ...
Linda T's user avatar
  • 101
5 votes
0 answers
134 views

Representing some odd multiples of integral homology classes by embedded submanifolds

Consider an $m$-dimensional compact closed orientable smooth manifold $M$ and an $n$-dimensional integral homology class $[\Sigma]$ on $M$, with $1 \le n \le m-1$. Then does there exist an odd integer ...
Zhenhua Liu's user avatar
2 votes
0 answers
195 views

Smooth compactification of complex varieties and uniqueness

Since I'm working in differential geometry, for the following I'm strictly interested in the smooth setting over $\mathbb{C}$ and its relation to the setting over $\mathbb{R}$. Here are a few useful ...
Paul Cusson's user avatar
  • 1,735
13 votes
0 answers
321 views

Nonsmoothable 4-manifolds

Does there exist a closed connected nonsmoothable 4-manifold $M$ such that: $\kappa(M)=0$ (Kirby-Siebenmann invariant vanishes, hence, there is no "classical" obstruction to smoothability) ...
Moishe Kohan's user avatar
  • 9,664
13 votes
1 answer
472 views

Impossibility of realizing codimension 1 homology classes by embedded non-orientable hypersurfaces

Suppose we have an $n+1$-dimensional compact closed oriented manifold $M$ and an $n$-dimensional integral homology class $[\Sigma]\in H_n(M,\mathbb{Z})$ on $M.$ Then is it true that $[\Sigma]$ mod $2$ ...
Zhenhua Liu's user avatar
3 votes
0 answers
183 views

Reference for a folklore theorem about h-cobordisms

I've seen referenced here that if $M$ and $N$ are closed topological $n$-manifolds and $f: \mathbb{R}\times M \to \mathbb{R}\times N$ is a homeomorphism, then $M$ and $N$ are h-cobordant. I know that ...
nick5435's user avatar
12 votes
1 answer
614 views

Isotopic diffeomorphisms of the sphere

Assume that $f:\mathbb{S}^n\to\mathbb{S}^n$ is a diffeomorphism and assume that there is an orientation preserving diffeomorphism $F:\mathbb{R}^{n+1}\to\mathbb{R}^{n+1}$ such that $F|_{\mathbb{S}^n}=f$...
Piotr Hajlasz's user avatar
2 votes
0 answers
127 views

Stable homeomorphism theorem for bi-Lipschitz mappings

The stable homeomorphism theorem says that: Every orientation preserving and surjective homeomorphism $f:\mathbb{R}^n\to\mathbb{R}^n$ can be written as $f=f_1\circ\ldots\circ f_k$, where $f_i:\mathbb{...
Piotr Hajlasz's user avatar
1 vote
0 answers
80 views

$L^p$-compression of metabelian groups

Question: Is there a metabelian group, so that for some $\epsilon >0$ and all $p \in [1, \infty[$ the [equivariant] compression exponent in [any] $L^p$-space is bounded by $1-\epsilon$ (bound does ...
ARG's user avatar
  • 4,342
5 votes
1 answer
321 views

Lattice generated by parabolics

Let $G$ be a semisimple Lie group of split-rank one and let $\Gamma$ be a non-cocompact lattice which is torsion-free. For the group $G=\mathrm{SL}_2(\mathbb{R})$ it then follows that $\Gamma$ is ...
Nandor's user avatar
  • 289
2 votes
0 answers
171 views

Stable homeomorphism theorem and the annulus theorem

Brown and Gluck [BG] proved in 1964 that the stable homeomorphism conjecture implies the annulus conjecture. Is the proof of this implication difficult? Is there any other place with the proof of ...
Piotr Hajlasz's user avatar

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