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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14
votes
0answers
176 views

Hauptvermutung for non-manifolds

The Hauptvermutung proposes the following: if two finite simplicial complexes are homeomorphic then they are PL-homeomorphic, meaning that they have a common refinement. People are mostly interested ...
9
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0answers
231 views

Finite quotients of surface braid groups

Let $\Sigma_b$ be a closed orientable surface of genus $b \geq 2$, and denote by $\mathsf{P}_2(\Sigma_b)$ the pure braid group with two strands on $\Sigma_b$. There is a braid $A_{12} \in \Sigma_b$ ...
9
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2answers
665 views

A plausible hyperbolic link

This link is hyperbolic according to SnapPy's computation. There is an obvious non-boundary parallel annulus spanned by two components at the very top in the diagram. If this annulus is essential, ...
3
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1answer
193 views

survey paper on the construction of hyperbolic manifolds

Is there a good survey paper which discusses the common ways of building hyperbolic $n$-manifolds?
7
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1answer
273 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 ...
2
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0answers
176 views

Can a non-compact manifold become compact by cutting it?

I'm trying to understand a step in a proof, where one starts with a non-compact manifold $V$ containing a trapped (2-sided, closed) surface $\Sigma$ that's non-separating. In order to complete the ...
2
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0answers
103 views

Direct limits of compact surfaces with uniformly bounded topology

Suppose we have a directed system of inclusions of compact surfaces with boundary $$S_1 \hookrightarrow S_2 \hookrightarrow S_3 \cdots $$ such that all of the surfaces $\{S_k\}$ have uniformly bounded ...
11
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2answers
578 views

Conjugacy classes of the mapping class group

I am not sure if this is a well known problem, but I was not able to find anything online that answered my question. Is it known how to tell whether two elements of the mapping class group of a ...
3
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1answer
96 views

Arcs and elements of the mapping class group

Is there any way to represent every element of the mapping class group of a surface as an arc on that surface?
4
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1answer
171 views

Universal cover or Bass-Serre tree: difference between definitions given by Bass and Serre

Let $(\mathbb G,\Gamma)$ be a graph of groups. A $G$-path from $u_0$ to $u$ is $$g_0e_1g_1\cdots e_{n}g_n,$$ where $e_1\cdots e_{n}$ is a walk in $\Gamma$ from $u_0$ to $u$ and each $g_i\in G_{s(e_i)}$...
6
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2answers
399 views

Topology/geometry of $O(2n)/U(n)$

I am interested in what is known about the topology (diffeomorphism type) or geometry (is it complex? non-complex? symplectic?) of either the compact space of orthonormal complex structures on $\...
20
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0answers
469 views

Is the mapping class group of $\Bbb{CP}^n$ known?

In his paper "Concordance spaces, higher simple homotopy theory, and applications", Hatcher calcuates the smooth, PL, and topological mapping class groups of the $n$-torus $T^n$. This requires an ...
4
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2answers
171 views

Rational surgery and attaching $2$-handles

It is well-known fact that integral Dehn surgeries on $3$-sphere $S^3$ are viewed as the result on the boundary of attaching $2$-handles $B^2 \times B^2$ to the $4$-ball $B^4$. Is there an analogue ...
2
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1answer
153 views

What is the Teichmuller metric on the Teichmuller space of a closed surface of genus 1?

Howard Masur's research asserts that if $S_g$ is a closed surface of genus $g\geq2$, then the Teichmuller space $T(S_g)$ does not have nonpositive curvature. His proof relies on the existence of ...
4
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1answer
199 views

Real points on a projective curve

Let $C_d$ be a smooth curve of degree $d$ in $\mathbb{CP}^2$. If we pick some homogeneous coordinates $[z_0:z_1:z_2]$ on $\mathbb{CP}^2$, then $C_d$ is the zero set of a generic polynomial of degree $...
7
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0answers
129 views

Representations of $\mathbb Z^2$ in ${\rm Symp}(S^2)$

Suppose $f_1$ and $f_2$ are two commuting symplectomorphisms of the sphere $\mathbb S^2$, of orders different from $2$. Is it possible to deform the pair $(f_1,f_2)$ to the pair of identity maps via a ...
5
votes
1answer
208 views

Riemann-Hurwitz for real maps

Let $S$ be a (compact, connected) Riemann surface of genus $g$ and $f: S\to \mathbb CP^1$ be a degree $d$ meromorphic function. Then the Riemann-Hurwitz formula tells us that the number of ...
4
votes
1answer
229 views

Growth rates of surface groups

I'm looking for readable references on calculating the growth rates of surface groups. There's some approach done briefly in page 159 of de la Harpe's "Topics in Geometric Group Theory", who cites: ...
2
votes
1answer
155 views

Hyperbolic links

Let $L\subset L'\in S^3$ be two links such that $L$ has one less number of components than $L'$. Further, $L$ is hyperbolic. Under what conditions is the link $L'$ hyperbolic. To be more specific $L, ...
5
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0answers
88 views

Cycles in Tits building

Tits building for an $n$-dimensional vector space $V$ is defined to be the simplicial complex corresponding to the poset of proper and non-zero subspaces of $V$. It is denoted by $T(V)$. This is known ...
3
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0answers
68 views

Unitary representations of Kähler groups deformable to one another as complex representations

Let $G=\pi_1(M)$ be the fundamental group of a compact Kähler manifold. Let $n$ be a non-negative integer. Then the set of homomorphisms $G\to GL(n, \mathbb{C})$ can be considered as a real variety $X$...
2
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0answers
43 views

Partitions of unity with arbitrary Lip-constants

Lets make things simple. Suppose we have a compact metric space $(X,d)$ and then some Lipschitz partition of unity exists, say a collection $\mathcal{F}=\{f_n\}$ subordinate to some open cover $\...
2
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0answers
138 views

boundary connect sum of Ganatra-Pardon-Shende

In Section 3.4 of https://arxiv.org/pdf/1809.03427.pdf, Ganatra-Pardon-Shende define the boundary connnect sum of two exact conical Lagrangians in a Liouville domain. In particular, in Figure 10, they ...
15
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1answer
579 views

Study topology from existence of multiple smooth structures?

There are classical existence results of a smooth structure on a topological manifold, and many results on the existence of multiple (i.e. exotic) smooth structures. Some utilize Freedman's theorem in ...
3
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1answer
151 views

Asymmetry of outer space - injectivity radius

I'd like to ask a question on "Asymmetry of Outer Space" by Yael Algom-Kfir & Mladen Bestvina. In Example $2$, page $4$ it says "Note that in this case the asymmetry can be explained by the fact ...
5
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0answers
71 views

Piecewise linear vs smooth high dimensional knots

A knot to me is the image of a smooth (pl locally flat) embedding $S^n \to S^m$ under the equivalence of smooth (ambient pl) isotopies. And more generally embeddings of closed manifolds $N \to M$. ...
2
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0answers
54 views

Cohomological dimension of closed $G$-invariant subspaces on homology manifolds with a group action

Suppose $G$ is a compact topological group acting on an $m$-homology manifold $M$ over some ring $R$ by homeomorphisms. Assume that the action of $G$ is effectively finite on a closed $...
4
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0answers
139 views

Symplectic isotopy problem for degree 3 curves in $\mathbb{CP}^2$

I am stuck in an argument in a paper written by Jean-Claude Sikorav. In the last section of his paper, he solved the symplectic isotopy problem for degree $3$ curves in $(\mathbb{CP}^2, \omega_{FS})$, ...
1
vote
1answer
91 views

Example incompressible branched surfaces

My study has led me to (incompressible) branched surfaces as described in https://core.ac.uk/download/pdf/82332579.pdf In the paper, the authors provide a great example, however I'm looking for more ...
9
votes
3answers
398 views

Does the mean curvature flow naturally come with less applications than intrinsic curvature flows?

I know studying the mean curvature flow is a very interesting area of research, I've fooled around with it a bit myself. But it honestly doesn't look like it has much applications within mathematics ...
3
votes
0answers
298 views

Is every manifold a double coset space?

Given any manifold $ M $ does there exist $ G $ a Lie group and $ H,\Gamma $ closed subgroups of $ G $ such that $$ M \cong \Gamma \backslash G/H $$ I was inspired to ask by this question: Example ...
15
votes
3answers
1k views

Unstable manifolds of a Morse function give a CW complex

A coauthor of mine and I want to use the following innocent looking statement in a forthcoming paper: Statement. Let $M^{2n}$ be a compact manifold and let $f$ be a Morse function with critical ...
3
votes
1answer
70 views

Number of curves in an admissible system of Jordan curves on a surface

Consider a compact Riemann surface of genus $g\geq2$. An admissible system of Jordan curves is a finite collection of Jordan curves $\{\gamma_1,\cdots,\gamma_n\}$ such that they are nonintersecting ...
6
votes
1answer
159 views

Non-example for Whitney (a) stratifications

Given a $C^1$ stratification $\mathscr{S}$ of a $C^1$ manifold $M$, we write $N^\ast \mathscr{S}$ for the union of conormals to the strata. The stratification is said to be Whitney (a) if $N^\ast \...
5
votes
1answer
170 views

Computation of \tau invariant

I am trying to understand the following inequality, $$0 \leq \tau (K_{+}) - \tau(K_{-}) \leq 1$$ from the following paper by Livingston. \ https://arxiv.org/pdf/math/0311036.pdf . At page 737 , he ...
10
votes
1answer
614 views

Searching for a Thurston paper with egg / 3-manifold analogy?

I remember coming across some article of Bill Thurston’s where he describes a 3-manifold (with boundary?) as being like an egg. In my recollection the interior of the egg, the shell, and even the ...
11
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0answers
241 views

Proving a group with two generators is not free that uses the Brahamagupta-Pell equation

Hello I encountered the following while reading a set of notes on free groups. It's not a homework question. "Does there exist a rational number $\alpha$ with $0 <|\alpha| < 2$ such that the ...
19
votes
1answer
642 views

Which cohomology classes are detected by tori?

Given a space $X$, I am looking for a characterization of classes $\alpha \in H^n(X;\bf Q)$ such that there is a map $f\colon T^n \to X$ so that $f^{\ast} \alpha$ pairs non-trivially against the ...
9
votes
1answer
274 views

Genus 2 3-manifolds bounding only $X^4$ with $b_2(X^4)$ big?

The genus of a closed orientable 3-manifold $M^3$ is the minimum genus among all Heegaard splitting surfaces for $M$. Every such 3-manifold bounds a compact 4-manifold. Let $I(M)$ denote the ...
6
votes
0answers
135 views

Coarsifying persistence modules

The context Let $I=[0,∞)$ and consider the category of persistence modules $(V,π)$ indexed over $I$ satisfying: For all $t$ in $I$ but a closed discrete set of points $T$, there exists a ...
5
votes
1answer
260 views

Proofs of Euler's characteristic formula for n-Dim polytopes

Twenty proofs of Euler's formula V - E + F - 1 = 1, which applies to convex polyhedrons, i.e., 3-dimensional polytopes, are presented at the Geometry Junkyard. I'm interested in proofs of the more ...
16
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0answers
594 views

Are these local systems on $\mathscr{M}_{g,1}$ motivic?

Let $(\Sigma_g, x)$ be a pointed topological surface of genus $g$, and let $MCG(g,1)$ be the mapping class group of this pointed surface. Then $MCG(g,1)$ has a natural action on $\pi_1(\Sigma_g, x)$ $$...
3
votes
1answer
122 views

Homogeneous manifold deformation retracts onto compact submanifold

Let $G$ be a connected Lie group. Then by a theorem of Cartan there is a diffeomorphism $$ G \cong K \times \mathbb{R}^n $$ where $K$ is a maximal compact subgroup of $G$. Now, let $M$ be a ...
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0answers
52 views

Boundary of slice disk exterior is the zero surgery of slice knot

I couldn't exactly guess the level of question. I asked in Math Stack Exchange. (Depending on the situation, I will remove it from here.) I'm trying to understand a sketch of proof of Livingston and ...
0
votes
1answer
149 views

Fundamental group of the complement of cell subcomplexes

Given a regular CW complex stucture on a manifold $C$ of dimension $n$ and a subcomplex $D$ of dimension $n-2$, I want to compute the fundamental group of the complement $\pi_1(C\setminus D)$. A ...
0
votes
0answers
91 views

Tubular neighbourhoods are unique up to ambient isotopy?

Let $M$ be a closed smooth submanifold of $N$. It is well known that tubular neigbourhoods of $M$ are diffeomorphic to the normal bundle of $M$ in $N$ and therefore to each other. Are they smoothly ...
3
votes
1answer
130 views

Two paths to the boundary with no holes in between

Let $X\subset \mathbb{R}^2$ be open connected (and let's say bounded), let $x\in X$ and $y\in\partial X$ be such that there is a Jordan curve $\gamma:[0,1]\to X\cup\{y\}$ such that $\gamma(0)=x$ and $\...
7
votes
2answers
245 views

What does the free action of a surface group on an R-tree look like?

Morgan and Shalen "Free action of surface groups on R-trees" 1989 shows that surface groups (genus at least 2) act freely on some real trees (R-trees). Their proof seems to be non-constructive, ...
3
votes
2answers
186 views

A question about congruence subgroups [closed]

For which $N_1$ and $N_2$ and $N$ be the greatest common divisor of $N_1$ and $N_2$, it is true that a congruence subgroup in $\mathrm{SL}_2(\mathbb{Z})$ generated by $\Gamma(N_1)\cup\Gamma(N_2)$ ...
7
votes
1answer
452 views

Classification of fibrations $\Bbb S^k\longrightarrow\Bbb S^d\longrightarrow B$

Does there exist a complete classification of all fiber bundles $\Bbb S^k\longrightarrow\Bbb S^d\longrightarrow B$, that is, fibrations of $\smash{\Bbb S^d}$ with each fiber homeomorphic to $\smash{\...

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