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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Does every hyperbolic, almost-transitive, triangulation of $\mathbb{R}^n$ have boundary homeomorphic with $\mathbb{S}^{n-1}$?

Question 1: Let $T$ be a triangulation of $\mathbb{R}^n$. Suppose that the 1-skeleton of $T$ endowed with the graph-metric (i.e. each 1-cell is given length 1) is Gromov-hyperbolic. Suppose moreover ...
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5 votes
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The figure eight knot complement in $S^3$

Recently I have been going through the book Hyperbolic Knot Theory by Jessica Purcell. In this book, there is an exercise in chapter 5, section 5.6, and exercise number 5.4 (Page - 101). This exercise ...
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4 votes
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132 views

Homotopy type of space of embeddings of a disk

Let $M^n$ be a smooth $n$-dimensional manifold and let $\mathbb{D}^n$ be the open unit disk in $\mathbb{R}^n$. Consider the space $\text{Emb}(\mathbb{D}^n,M^n)$ of smooth embeddings of $\mathbb{D}^n$ ...
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7 votes
1 answer
142 views

A ribbon presentation for a torus knot

Let $K$ be a knot in $S^3$. It is well-known that the knot $K \# -\overline{K}$ is always ribbon. The following picture describes the connected sum of the left-handed torus knot $T(3,4)$ and the right-...
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3 votes
3 answers
188 views

Reference for an easy lemma on homeomorphisms of connected manifolds

If M is a connected manifold of dimension $\geq 2$ then the set of orientation preserving homeomorphisms of M that are isotopic to the identity acts $n$-transitively on M for all positive $n\in\mathbb{...
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2 votes
1 answer
48 views

fills a given polygon with a few types of given primitives

Given one large 2D polygon, and K types of small polygons (the primitives). For each type of small polygon, it can be rotated, and has an infinite number of pieces. For such a Jigsaw puzzle game, is ...
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10 votes
1 answer
436 views

Does a compact contractible metric space have a point that is fixed by all isometries?

Let $(X,d)$ be a compact and contractible metric space. Let $\operatorname{Isom}(X)=\{\phi\colon X\to X\}$ be its group of isometries. Question: Is there a point $x\in X$ fixed by all $\phi\in\...
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8 votes
2 answers
248 views

Symmetries of contractable subsets of $\Bbb R^n$

Let $K\subset\Bbb R^n$ be a non-empty compact subset of $\Bbb R^n$. A symmetry of $K$ is an isometry of $\Bbb R^n$ that fixes $K$ set-wise. Since $K$ is compact, there is always a point $x\in\Bbb R^n$ ...
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3 votes
0 answers
145 views

Can Whitehead manifold admit a properly discontinuous cocompact group action?

Can classical contractible manifolds such as Whitehead manifold admit a properly discontinuous cocompact group action? Here "properly discontinuous" doesn't have to be fixed point free, but ...
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7 votes
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On exotic symplectic structures of smooth closed 4-manifolds

What are some known techniques and examples of exotic symplectic structures on a fixed smooth closed 4-manifolds [by exotic I mean two symplectic structures that are not symplectomorphic]. This sounds ...
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0 votes
0 answers
71 views

Inverse limit in category of graphs

Let $... \leftarrow G_i \leftarrow G_{i+1} \leftarrow ...$ be an inverse system of graphs (which might not be locally finite), morphisms are symplicial maps (we allow collapsing of edges to vertices)....
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6 votes
2 answers
290 views

Is there any genus one knot other than $5_2$ with Alexander polynomial $2t^2-3t+2$?

It is known that genus one fibred knots are two trefoils and the figure-eight knot. Is there any characterization of the knot $5_2$? Specifically, is there any other genus one knot that shares the ...
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2 votes
1 answer
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Is every finitely generated classical Schottky group quasifuchsian?

$\DeclareMathOperator\PSL{PSL}$(Classical, finitely generated) Schottky groups are groups generated by finitely many hyperbolic elements of $A_i\in \PSL(2,\mathbb{C}), $ $i<n$ such that the ...
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2 votes
0 answers
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Connection between a function and its usage in geometry [closed]

I know nothing about geometry, but I found a function which seems to have something to do with geometry. This function is, $$f(x,y,z) = \dfrac{(x,y,z)}{\sqrt{1 + x^2 + y^2 + z^2}}$$ where $x,y,z$ is ...
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11 votes
1 answer
290 views

How wild can an open topological 3-manifold be if it has a compact quotient?

Let $M$ be an open, simply connected, 3-manifold. Suppose $M$ admits a properly discontinuous, co-compact topological action by a finitely generated group. Question 1: If $M$ is 1-ended, must it be ...
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9 votes
1 answer
198 views

Is there a geometric interpretation of the second derivative of the Alexander polynomial at $1$?

For an (oriented) knot in $S^3$ the number $\Gamma(K) := \Delta_K’’(1)$ shows up in a number of places in knot theory, for example the Casson-Walker-Lescop invariant. Here $\Delta_K(t)$ is the ...
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3 votes
0 answers
92 views

Survey or good reference of taut foliations

I am interested in the topology of foliations. In particular, I want to understand taut foliations, or projectively Anosov flows, and Anosov flows. I guess that A. Candel and L. Conlon, Foliations I (...
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5 votes
1 answer
174 views

Coefficient of the top Pontryagin class in $L$-genus

The $L$ genus can be expressed as combinations of the Pontryagin classes with the first few terms as follows: $$L_1=\frac{1}{3}p_1,$$ $$L_2=\frac{1}{45}(7p_2-p_1^2),$$ $$L_3=\frac{1}{945}(62p_3-...
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3 votes
0 answers
74 views

Intersection number for 4 manifold with boundary

Let $X$ be a closed oriented smooth $4$-manifold. Suppose there is an embedding $\Sigma\to X$, it is known that the self-intersection number satisfies $[\Sigma]\cdot [\Sigma]=\pm\int_\Sigma c_1(N)$, ...
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3 votes
1 answer
93 views

Complex length of geodesic added in hyperbolic Dehn surgery

Suppose $M$ is a cusped finite-volume hyperbolic $3$-manifold, say with a single cusp for simplicity. Following [NZ, Section 4] we can parametrize deformations of the hyperbolic structure with a ...
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1 vote
1 answer
59 views

Are two nonsimple closed geodesics in minimal position?

We know that two simple closed geodesics are in minimal position, meaning that they realize the geometric intersection number. Is this result true for a pair of nonsimple closed geodesics?
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2 votes
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Questions about symmetric spaces

I'm a little confused with the following questions: (1) Why does a symmetric space $M=G/K$ of compact type have $\mathcal{R}^M\geq 0$? (2) Moreover, why does $\chi(M)\neq 0$ if and only if ${\rm rk}\ ...
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1 vote
0 answers
134 views

Fiber bundle orientability vs manifold orientability

This question seems like a pretty straightforward generalization of a result from vector bundles but its been on MSE for over a week with no answers so I'm reposting https://math.stackexchange.com/...
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6 votes
0 answers
111 views

Smoothing tame topological knots, from an analytic perspective

A tame topological knot (for the purpose of this question) is a topological embedding of $S^1 \times D^2$ into $\Bbb R^3$. Tame topological knots are known to be isotopic to smooth knots. This ...
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1 vote
0 answers
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Do all manifolds admit metrics with Euclidean balls?

Let $M$ be a compact topological n-manifold. Suppose we are given a locally flat embedding $M \subset \mathbb{R}^{n+k}$. This induces a metric on $M$ by restriction. Is it true that for $\epsilon$ ...
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4 votes
1 answer
78 views

Does length spectrum determine a hyperbolic 3-manifold? What if we also know holonomies?

Question 1: Suppose that two hyperbolic 3-manifolds $M_1$ and $M_2$ with finitely generated fundamental group satisfy the property that for every closed geodesic in $M_1$, there is a closed geodesic ...
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6 votes
1 answer
267 views

Does every simply connected, orientable, non-compact, 3-manifold embed in $\mathbb{R}^3$?

Let $M$ be a simply connected, (orientable), non-compact, 3-manifold without boundary. Must $M$ be homeomorphic with a topological subspace of $\mathbb{R}^3$?
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3 votes
1 answer
52 views

Given a Heegaard splitting $M = V\cup_F W$, then $V\setminus N(D_1)$ is ambient isotopic to $V\cup N(D_2)$ for a meridian pair $\{D_1,D_2\}$

I sincerely apologize if MathOverflow is not the appropriate place to ask this question. I also tried consulting M.SE but it seems that this question gained little to no interest . Consider a ...
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9 votes
3 answers
297 views

Is it possible that every edge in a 1-planar drawing with minimum number of crossings is crossed?

A graph is 1-planar is it has drawing in the plane so that each edge is crossed at most once. Here we also assume the drawing satisfies (1) no edge is self-crossed; (2) no two adjacent edges are ...
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8 votes
1 answer
264 views

Surgery along knots and connected sum

Denote $S^3_{p/q}(K)$ by performing $p/q$-surgery along a knot $K$ in $S^3$. Let $K$ and $J$ be two arbitrary oriented non-trivial knots in $S^3$. Is there a nice relation between surgery on the ...
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14 votes
2 answers
691 views

Must a space that is locally injective image of $\mathbb{R}^n$ be a manifold?

Suppose $X\subseteq\mathbb{R}^m$ s.t. for any $x\in X$ and any open $U\subseteq\mathbb{R}^m$ that contains $x$, there exists a smaller open set $V\subseteq U$ also containing $x$, so that $V\cap X$ is ...
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3 votes
1 answer
229 views

“Combinatorial” moves between cell complexes

EDITED: A pair of finite simplical complexes are equivalent if and only if they are related by a finite sequence of the Pachner moves. Is there a similar thing on finite cell complexes? That is, are ...
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11 votes
0 answers
310 views

Is my open set a ball?

Let $P$ be a polytope of dimension $n$, and let $\mathcal{C}$ be a polyhedral subdivision of $P$. That means that $\mathcal{C}$ is a finite collection of polytopes whose union is $P$, such that the ...
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2 votes
0 answers
98 views

Is each good topological 3-orbifold homeomorphic to a CW-complex?

Suppose a group $\Gamma$ acts properly discontinuously by homeomorphisms on a 3-manifold $M$. Thurston observed that $M/\Gamma$ has the structure of a (topological) orbifold. (This generalises to any ...
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8 votes
0 answers
271 views

Fast method to verify if a point belongs to a given convex $d$-polytope

We are given a $d$-dimensional convex polytope $P\in\mathbb{R}^d$. Assume we have all the supporting hyperplanes describing $P$ and its vertices. Let $S$ be a sequence of $n\gg 1$ points $\mathbb{R}^d$...
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8 votes
0 answers
217 views

A conjecture about homotopy $S^1\times B^3$'s

$\textbf{Conjecture}:$ Let $X^4$ be a homotopy $S^1\times B^3$ with the following properties: Attaching a four dimensional 2-handle gives a standard $B^4$. The $k$-fold cyclic cover is diffeomorphic ...
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13 votes
1 answer
245 views

Mapping torus of Klein bottle

This got 5 upvotes but no answers on MSE (Mapping torus of Klein bottle), so I'm cross-posting to MO: The mapping torus of a Klein bottle $ K $ is a compact flat 3 manifold. The mapping class group of ...
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6 votes
1 answer
94 views

Positive vs negative Dehn twist monoids

Given a bounded surface $S$ and a mapping class $h$, construct the open book 3-manifold $(S,h)$. If $h$ lies in the positive monoid of the mapping class group, then $(S,h)$ supports a tight contact ...
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8 votes
1 answer
331 views

Universal cover with one end

Suppose that $M$ is a non-compact manifold of finite topological type with one end which is the universal cover of some closed manifold $N$. Is $M $ necessarily homeomorphic to the total space of some ...
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  • 6,338
8 votes
1 answer
119 views

Isomorphism type of the knot concordance group

Isotopy classes of oriented knots in $S^3$ form a commutative monoid with respect to connected sum. Smoothly slice knots, i.e. knots that are the boundary of a smooth properly embedded disk in $B^4$, ...
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29 votes
1 answer
1k views

Are homeomorphic representations isomorphic?

Let $G$ be a finite group. Let $V_1, V_2$ be two finite-dimensional real representations. Suppose $f: V_1 \to V_2$ is a $G$-equivariant homeomorphism. Can one conclude that $V_1$ and $V_2$ are ...
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10 votes
1 answer
153 views

The knot $K\subset \Bbb S^3$ is smoothly slice, but the disc $D\subset \Bbb D^4$ is only locally flat. Can $D$ be smoothed?

Suppose I am given a smoothly slice knot $K\subset\Bbb S^3$. But I am only given a locally flat disc $D\subset \Bbb D^4$ with boundary $K$. Question: Is there a smooth disc $D'\subset\Bbb D^4$ with ...
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  • 9,471
11 votes
1 answer
417 views

Counterexamples to an analog of Cannon's Conjecture which do not arise from manifolds?

Let $\Gamma$ be a finitely presented hyperbolic group with boundary homeomorphic to $S^{n-1}$. Are there any examples of such $\Gamma$ which are known to not be the fundamental group of any $n$-...
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  • 255
5 votes
1 answer
188 views

Nondegeneracy of kernel of map on homology induced by covering of surfaces

Let $f:X \rightarrow Y$ be a finite covering map between compact oriented surfaces and let $K$ be the kernel of the induced map $f_\ast: H_1(X) \rightarrow H_1(Y)$. Here homology has rational ...
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  • 53
2 votes
1 answer
112 views

Mapping class group and pure mapping class group (and their generating sets)

$\DeclareMathOperator\Mod{Mod}\DeclareMathOperator\PMod{PMod}\DeclareMathOperator\Homeo{Homeo}$I am very confused about the definition of mapping class group and pure mapping class group (and their ...
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6 votes
1 answer
197 views

A (possible) generalization of the unknot, inverses and the knot concordance

The notion of a knot concordance is a rich subject in low-dimensional topology, see Livingston's survey. More precisely: For $i=0,1$, let $K_i$ be knots in $S^3$. A knot concordance from $K_0$ to $K_1$...
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21 votes
1 answer
630 views

Is $\mathbb{CP}^3$ minus two points the universal cover of a compact manifold?

After reading some recent questions on mathoverflow about universal coverings, I am curious about the following: Is it possible to construct a closed $6$-manifold $M$, with universal cover ...
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14 votes
4 answers
1k views

A notion of 2-dimensional tree

Summary: This post has got rather long after the discussion. The main still open Questions are 5 & 6 below. There is work in progress, and I'll post an update at some point. A tree is a connected ...
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3 votes
0 answers
131 views

What is the meaning of local inertia conjugation property?

In Hatcher, Allen; Lochak, Pierre; Schneps, Leila, On the Teichmüller tower of mapping class groups, J. Reine Angew. Math. 521, 1-24 (2000). ZBL0953.20030., we have: Abstract. Let $\widehat{G T}^{1}$ ...
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1 vote
1 answer
88 views

Transversal pre-image of a small enough trivial tubular neighborhood contains a trivial tubular neighborhood

A similar post on MSE without answer. Let $f\colon M'\to M$ be a smooth map between two orientable closed smooth manifolds and $S$ be a smoothly embedded closed orientable submanifold of $M$ of co-...
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