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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Computation on Characteristic classes

I am organizing a reading seminar on Characteristic Class. The audience in the seminar is interested Symplectic and Contact manifold. I work in Categorification and would like to compute some ...
2 votes
0 answers
12 views

Translation length on annular curve graphs

Question about curve stabilisers acting on annular curve graphs, plus context since I'm interested in being fact-checked. Definition: let the group $G$ act by isometries on a metric space $(X,d)$. ...
7 votes
1 answer
186 views

Extending diffeomorphisms

Suppose we have a diffeomorphism $f:{\mathbb{S}}^n_{+}\to\mathbb{S}^n$ of class $C^1$ of the closed upper hemisphere onto a submanifold of $\mathbb{S}^n$ with boundary. Question. Is it possible to ...
3 votes
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37 views

Bounding the Betti numbers of Čech complexes in Euclidean space

Let $S$ be a set of $n$ points in $\mathbb{R}^d$, where $d \ge 2$. Then let $C=C(S)$ denote the union of closed balls of radius $1$ centered at points of $S$. For $0 \le j \le d-1$, how large can the ...
3 votes
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103 views

Diffeomorphism problem for complex surfaces?

I'm sure the following is well known by the right people, I'm just hoping for some pointers. I know about Markov's theorem that the diffeomorphism problem for general 4-manifolds is undecidable. Let $...
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2 votes
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Cubic lattice representation of a solid torus knot using the surface as a boundary

For physics simulation reasons, I would like to respresent a solid torus knot as a collection of integer points sat on a cubic lattice. If I were to do this using a sphere, I would do this by saying ...
6 votes
1 answer
174 views

In knot theory, what is this link property and how to detect it: "linkings between components separate nicely"

The following could be made more general (see below), but let's focus on a link $L$ that consists of three components (closed curves) $\gamma_1,\gamma_2,\gamma_3\subset\Bbb R^3$. Call $L$ a necklace ...
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4 votes
2 answers
182 views

Books for learning branched coverings

I am self-studying branched coverings. I read it from B. Maskit's Kleinian groups book. I want some more references for reading branched covers. In particular, I want to understand how to create ...
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5 votes
1 answer
331 views

Six people standing on earth

Consider 6 people $p_i$, $i=1,\dots 6$, standing on a sphere $S^2$. We label the positions of these people by $p_i$ again. Suppose no pair of these points $p_i$ are antipodal. At each point $p_i$ ...
5 votes
1 answer
222 views

Are two slice surfaces with minimal genus isotopic?

For a link $L\subset S^3$ and two Seifert surfaces (edit: a better name would be slice surfaces as the comments below 1 2 point out) with minimal genus $S_1,S_2\subset B^4$, I have the following ...
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7 votes
1 answer
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Why does the tangent classifier classify the tangent (micro)bundle?

Let $\mathcal{M}\mathrm{fld}_n$ denote the $\infty$-category of topological manifolds (without boundary) and embeddings; more precisely, it is the homotopy coherent nerve of the simplicial category ...
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6 votes
1 answer
219 views

Proper action on product manifold

Suppose that $\mathbb{R}^n$ is the maximal group that can act properly on a manifold $N$ and $\mathbb{R}^m$ is the maximal group that can act properly on a manifold $M$ ( i.e, $\mathbb{R}^{m+1}$ can't ...
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1 answer
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Space-time trajectory that cannot be straightened and its braid form

Considering we have the space-time trajectory of multiple particles (or any objects) in the X-Y-Time coordination system. Given a projection direction, we can obtain the braid form of the space-time ...
4 votes
1 answer
271 views

Faithful locally free circle actions on a torus must be free?

Do we have an example of a smooth action $S^1 \curvearrowright T^n$ which is faithful, locally free but not free? I know such an action must induce an injection $\rho:\pi_1(S^1)\to\pi_1(T^n)$. Another ...
14 votes
2 answers
768 views

Can a cyclic group of prime order act on a rationally acyclic finite dimensional complex and have no fixed points?

Let $C_p$ b the cyclic group of order $p$, with $p$ a prime. Is is possible for $C_p$ to act (cellularly) on a rationally acyclic finite dimensional CW complex $X$ with no fixed points? Standard Smith ...
3 votes
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Extending continuous maps from spheres to Euclidean spaces [migrated]

Fix $d\in\mathbb{N}$. Consider the following sets as topological spaces with the subspace topology from $\mathbb{R}^{d+1}$. $$S^d = \{ (x_0,\ldots,x_d)\in\mathbb{R}^{d+1}\mid \sum x_i^2 = 1\}$$ $$ D^{...
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1 answer
56 views

Sets with a good lift under a covering

Suppose I have a covering map $\pi : E \to X$ between (nice) topological spaces, and $x \in X$. If $U \ni x$ is a very small open set, then $\pi^{-1}(U)$ is a discrete union of subsets $V_d \subset X$ ...
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6 votes
0 answers
217 views

11/8-type inequality from Heegaard Floer theory?

Background: a well-known conjecture in $4$-dimensional topology states that if $M$ is a $4$-dimensional oriented closed spin smooth manifold, then the second Betti number of $M$ is bounded from below ...
1 vote
1 answer
154 views

Ways to prove that $n$-component Brunnian link is nontrivial

The attached image shows a way to construct an $n$-component Brunnian link for any $n\geq 3$. That is, this link is not trivial, but deleting any of its components makes the new link trivial. The ...
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9 votes
2 answers
402 views

Bing sling isotopy to unknot

Rolfsen asked the question as to whether any knot is topologically isotopic to the unknot. Where a topological isotopy is a continuous path in $\operatorname{Emb}(S^1,\mathbb{R}^3)$. From now on I ...
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A question from the paper "Hyperbolic rigidity of higher rank lattices" by Thomas Haettel

In the paper "Hyperbolic rigidity of higher rank lattices" by Thomas Haettel, the author has made the following statement in the proof of Corollary D in page no. 18 ( https://arxiv.org/pdf/...
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7 votes
1 answer
156 views

Lens space bounding a topological, simply-connected 4-manifold with $b_2=1$

The following is written in section 1.6 (p.7) of this paper: https://arxiv.org/pdf/1010.6257.pdf. ($\cdots$) Which lens spaces bound a smooth, simply-connected 4-manifold $W$ with $b_2(W)=1$? ($\cdots$...
2 votes
0 answers
157 views

Determinantal variety

It is well known in literature about the determinantal varieties, symmetric determinantal varities, skew-symmetric determinantal varieties. Is it possible to study determinantal varieties over the ...
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3 votes
1 answer
182 views

A Mazur manifold bounded by $\Sigma(2,3,13)$

Using Kirby calculus, Akbulut and Kirby first analyzed that the Brieskorn sphere $\Sigma(2,3,13)$ is diffeomorphic to the following link in their famous paper: Then they switched the circles when ...
2 votes
0 answers
97 views

Unstably dualizable maps

Call a map between compact, connected framed $n$-manifolds $f:M \rightarrow N$ unstably dualizable if there exists an $f':N \rightarrow M$ such that the following diagram commutes up to homotopy: $$\...
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8 votes
1 answer
247 views

Non-compact three-manifolds with the same proper homotopy type are homeomorphic?

I am looking for some literature with some (counter) examples of the following fact (though I don't know if the fact is true or not): Let $M, M'$ be two non-compact connected $3$-manifolds with the ...
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3 votes
1 answer
195 views

Picturing twisting of strands explicitly after blow downs

In order to simplify Kirby calculus proofs, one can use a box notation which indicates a number of full twists up to sign. In the scenario of two strands with twist boxes, it is straightforward to ...
1 vote
0 answers
84 views

Sufficient condition for moduli space of slope-stable bundles to be non-empty

I'm trying to find to a statement concerning the non-emptiness of the moduli space of slope stable vector bundles over a Kähler surface in the literature. Let $X$ be a Kähler surface. Let $\mathscr{M}(...
3 votes
1 answer
123 views

Example of homeomorphism that lifts to real blow up but not C^1?

Given smooth manifold $M$, let $Bl_\Delta(M\times M)$ be the (say oriented; you can ask this question for the unoriented case too) real blow up of $M\times M$ along the diagonal and let $\pi:Bl_\Delta(...
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5 votes
0 answers
229 views

A certain kind of proof of the Hairy Ball Theorem

I'd just like to know if proofs of the Hairy Ball Theorem along the following lines are well-known or even somewhere in the literature. From a given vector field $V_1$ on $S^2$, form another, $V_2$, ...
8 votes
1 answer
240 views

Topology of a smoothing of an isolated singularity

Let $(X,x)$ be an affine variety with a normal isolated singularity and $Y$ be a smoothing of $X$ (for this we mean $Y$ can be realized as a smooth fibre of a deformation of $X$). Question. Can we ...
7 votes
1 answer
179 views

If the number of ends of Freudenthal space is infinite, then its space of ends is homeomorphic to the Cantor set?

I don't know whether this is the right place to discuss a part of someone's thesis or not. If it is wrong, let me know; I will delete my post. I am reading this thesis. Corollary 4.1.15. on page 63 ...
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5 votes
0 answers
81 views

Standard 2-instantons on the 4-sphere under conformal transformation

It is well-known that there is a standard SU(2) 1-instanton on the 4-sphere and any 1-instanton can be obtained from the standard one by conformal transformation. Is there any explicit (family of) &...
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5 votes
0 answers
108 views

The fundamental group of the complement of badly embedded open $n$-ball in $\Bbb R^n$

Let $\mathcal D^n$ be an open subset of $\Bbb R^n$ such that $\mathcal D^n$ is homeomorphic to $\{x\in \Bbb R^n:|x|<1\}$. Suppose $\Bbb R^n\setminus \mathcal D^n$ is path-connected. How bad can $\...
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3 votes
0 answers
118 views

Algebraic variations of the full knot Floer complex

In Hom's paper (arXiv link), p.20, Section 3.3 ends with "There are other algebraic modifications one may consider, such as setting $U^n = 0$ or $UV = 0$", referring to the knot Floer ...
11 votes
2 answers
1k views

Is there a contractible hyperbolic 3-orbifold of finite volume?

Let $\mathbb{H}^3:=\operatorname{SO}(3,1)/\operatorname{O(3)}$. Is there a lattice $\Gamma$ in $\operatorname{SO}(3,1)$ such that \begin{equation} X:=\mathbb{H}^3/\Gamma \end{equation} is contractible?...
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5 votes
1 answer
241 views

Homology of spherical $3$-manifold group

I have been studying $3$-manifolds recently and I got stuck in the following situation. For lens spaces the below fact is true. Let $G$ be a finite group acting freely and orthogonally on $S^3$ so ...
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1 vote
0 answers
68 views

about codimension two foliation

Are there examples of codimension 2 foliations on closed compact 4-manifolds or 5-manifold I am curious about examples of codimension Are there any previous studies or lecture notes of foliation ...
10 votes
3 answers
461 views

Centre of orbifold fundamental group of torus (Klein bottle) with one cone point

$\newcommand{\orb}{\mathrm{orb}}$Let $T$ ($K$) be the torus (Klein bottle) with one cone point of order $q\geq 2$. The presentation of their orbifold fundamental groups are easy to find. Namely, $$\...
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0 votes
0 answers
48 views

Obstruction to finding a framing for quotient manifolds

The question is rather open-ended but I hope it is concrete enough. If $M$ be a closed parallelizable smooth manifold with a smooth properly discontinuous co-compact action of a Lie group $G,$ what ...
6 votes
1 answer
204 views

How to prove that Lie group framing on S^1 represents the Hopf map in framed cobordism

The Pontryagin-Thom construction gives an isomorphism from the stable homotopy groups of spheres and framed cobordism groups. It seems to be well-established that for dimension 1 (see this question), ...
3 votes
0 answers
56 views

Mostow-Palais equivariant embedding for manifolds with corners

Let $M$ be a compact smooth manifold and let $G$ be a connected compact Lie group acting on $M$. According to an old theorem of Mostow and Palais, there exists a $G$-equivariant embedding of $M$ into ...
6 votes
0 answers
60 views

Classification of contractible open n-manifolds which embed in a compact n-manifold

Does there exist a classification of contractible open $n$-manifolds ($n\geq 3$) which embed in a compact $n$-manifold? More general, does there exist a classification of contractible open $n$-...
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17 votes
1 answer
388 views

Topology of the space of embedded genus $g$ surfaces in $S^3$

Consider the space of smoothly embedded genus $g$ surfaces in the 3-sphere under the $C^\infty$ topology: $$\mathcal E_g:=\operatorname{Emb}(\Sigma_g,S^3)/{\operatorname{Diff}(\Sigma_g)}$$ where $\...
5 votes
3 answers
168 views

Ideal triangulations of $3$-manifolds with "cusps" of genus $\ge 2$

Typically when one thinks about ideal triangulations of a $3$-manifold the link of each ideal vertex is a circle, so the ideal points correspond to toroidal cusps; alternatively, one can truncate the ...
3 votes
1 answer
126 views

Existence of covering isomorphism

Let $C,D$ be two non-compact complex algebraic smooth curves. Suppose that two unramified regular finite maps $p_1, p_2: C \rightarrow D$ are given and have the same degree. Is there always an ...
5 votes
1 answer
203 views

Stable torus that is not a torus [duplicate]

Let $M$ be a closed manifold such that $M\times \mathbb{S}^1$ is a torus. Is it true that $M$ is homeomorphic to a torus?
6 votes
0 answers
286 views

Arithmetic Teichmüller curves, first eigenvalue of the Laplacian, McMullen's expander conjecture

$\DeclareMathOperator\SL{SL}$By a result due to Ellenberg and McReynolds, any finite index subgroup $\Gamma$ of $\Gamma(2) \subset \SL\left(2,\mathbb{Z}\right)$ is the Veech group of an arithmetic ...
3 votes
0 answers
232 views

Representations of triangle groups

$\DeclareMathOperator\SU{SU}\DeclareMathOperator\PSL{PSL}$I am self-studying triangle groups and the following question comes up. Let $G$ denotes $(2,3,7)$ triangle group. It is symmetry group of $(2,...
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5 votes
1 answer
154 views

Question about and good reference for Kahn and Markovic result

As far as I understand, the celebrated result of Kahn and Markovic about quasi-Fuchsian immersions of surfaces in hyperbolic 3-manifolds has the following corollary: Let $M$ be a compact hyperbolic $3$...

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