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).
4,407 questions
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Does Bungert–Slepčev rigidity imply Γ-convergence to Laplacian in relational graph frameworks? [closed]
Consider quasi-uniform point clouds in \mathbb{R}^3 with symmetric positive weights w_{ij}^\ell normalized as \sum_j w_{ij}^\ell = O(\ell^{-2}). Bungert–Slepčev (2025) prove rigidity: finite non-...
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0
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141
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Orientability of orbits
Are the orbits of an action by a simply connected Lie group on an orientable manifold necessarily orientable?
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5
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1
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168
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Periodic mapping classes with homeomorphic mapping tori
Let $\Sigma_g$ be the closed orientable surface of genus $g\ge 2$. Suppose $\phi_1$ and $\phi_2$ are periodic mapping classes on $\Sigma_g$ such that the mapping tori $M_{\phi_1}(\Sigma_g)$ and $M_{\...
- 345
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0
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91
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Homeomorphisms of Minkowski space
I was reading the paper “On two topologies that were suggested by Zeeman” by Papadopoulos and Papadopoulos. In Theorem 1.1, the authors show that the group of homeomorphisms of Minkowski space ...
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4
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0
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120
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Homological blocks as mathematical objects
Homological blocks, introduced by Gukov–Pei–Putrov–Vafa (GPPV), are conjectural q-series invariants of 3-manifolds expected to refine or categorify the Witten–Reshetikhin–Turaev invariants and exhibit ...
18
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2
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1k
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Proving that orientable surfaces are orientable with my hands tied behind my back
I'm writing a topology book aimed at fairly inexperienced students. I would like to describe the classification of surfaces in it. Since my audience does not know what a smooth manifold is (just a ...
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2
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0
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119
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Compactifications for the space of embeddings from a closed subset of the Cantor set to the sphere
Let $K$ be an infinite closed subset of the Cantor set and let $\mathrm{Emb}(K,\mathbb{S}^{2})$ be the space of embeddings of $K$ into $\mathbb{S}^{2}$ equipped with the compact-open topology. On $\...
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185
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The relation between Temperley-Lieb algebra and representations of $U_q \mathfrak{sl}_2$
What is a good modern reference (or maybe this is known to someone here and lacks a good reference) which explains the relation between the Temperley-Lieb algebra and representations of $U_q(\mathfrak{...
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2
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0
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201
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Bestvina-Brady Proposition 3.9
The following proposition and proof is given in Bestvina and Brady's "Morse theory and finiteness properties of groups".
Proposition 3.9. Let $H$ be a finitely presented group. Suppose that ...
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79
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Why does Kirby move of type 1 replace corresponding 4-manifolds $W$ by $W \mathbin\# \mathbb CP^2$ or $W \mathbin\# \overline{\mathbb CP^2}$?
I am studying Kirby Calculus from the book named “Lectures on the topology of $3$-manifolds” by Nikolai Saveliev .
I have read about two Kirby Moves. We know integral surgery on a link represents a $3$...
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118
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Holomorphic fiber bundles with holomorphic and non-holomorphic sections
I am looking for examples of holomorphic fiber bundles $\pi\colon E \rightarrow B$ with the following properties:
Both $E$ and $B$ are connected complex manifolds.
There exists a holomorphic section $...
- 1,438
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0
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37
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Prove rigorously a constraint on the efficiency of intermediate segments in a polygonal chain over a regular $k$-grid
Let $k, r \in \mathbb{N}$ with $k \ge 2$ and $r \ge 3$, and let $n_1, n_2, \ldots, n_k \in \mathbb{N}$ satisfy $3 \le n_1 \le n_2 \le \cdots \le n_k$.
We define
${G_n}^k := \{0,1,\ldots,n_1-1\} \times ...
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166
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On the existence of an embedding into $\mathcal S^7$
Let $X$ be a Calabi-Yau 3-fold considered as real compact 6-fold. Suppose that $Y$ inside $X$ is a singular compact complex curve.
Let $U$ be a smooth open neighborhood of $Y$ which is homotopy ...
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2
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1
answer
231
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Index of the hyperelliptic mapping class group’s image in the integral symplectic group
For $g\ge 3$, let $\rho:\mathrm{Mod}(S_g)\to \mathrm{Sp}(2g,\mathbb Z)$ be the symplectic representation, and let $\mathrm{SMod}(S_g) < \mathrm{Mod}(S_g)$ denote the hyperelliptic mapping class ...
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6
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1k
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Computationally inspired theorems and conjectures in knot theory
I am collecting examples of theorems and conjectures in knot theory that were originally discovered (or inspired) by computer experiments.
Examples include:
Hoste’s conjecture on zeros of the ...
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5
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1
answer
283
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Involution of a disk which fixes a point on the boundary
Question: Let $f\colon D^n\to D^n$ be a continuous involution which fixes a point $x\in \partial D^n$. For every open neighborhood $U\subset D^n$ of $x$, is there necessarily a fixed point of $f$ in $...
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0
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314
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Subtle gap between PL & SMOOTH in dimension 4
Main Question and Subtlety
The folk wisdom, often cited as a clear truth, states, "PL = SMOOTH in dimension 4". While intuitively appealing, this is not a precise mathematical statement. The ...
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4
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193
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Simplicial structure of outer space
Let $CV_n^r$ denote Culler-Vogtmann reduced outer space (graphs do not have separating edges) (see Moduli of graphs and automorphisms of free groups for reference), and let $G$ be a marked graph in ...
- 439
4
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130
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The Alexander polynomial of quasi-projective groups
Given a quasi-projective variety $M$, Dimca, Papadima, and Suciu (Theorem 4.3 and Corollary 4.4) showed that when (the first Betti number) $b_1(\pi_1(M))\ge3$, then the Alexander polynomial of $\pi_1(...
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1
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286
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Computing attaching maps in triangulation with SnapPy
Forgive me if this is basic, I am new in SnapPy. I want compute, for a given 3-manifold in SnapPy (ex. as running example M = Manifold('m004'), the figure-eight ...
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129
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Could any one explain why $CFK(K_-)$ can be viewed as a mapping cone?
I am currently reading Ozsváth and Szabó's paper On the skein exact sequence for knot Floer homology. I hope to understand what exactly are the maps in the skein exact sequence. In Section 3, they ...
6
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298
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Complement of Brownian motion
Consider the path of a 3d Brownian motion. Is its complement homeomorphic to some fixed $3$-manifold with probability one? If not what can we say about the complement? What about 4d?
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6
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2
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355
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On a closed hyperbolic surface, how do we know there exist infinite non-closed simple geodesics spiraling towards closed geodesics?
In Casson and Bleiler's Automorphisms of Surfaces after Nielsen and Thurston, they include a diagram
of an infinite non-closed simple geodesic on a closed hyperbolic surface, which limits on either ...
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0
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135
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Embedding smooth manifold in Euclidean space so it is easy to see it is a smooth retract of a neighborhood
I'm teaching smooth manifolds this semester. For various reasons, it would be very helpful if early in the course I could prove the following. Let $M$ be a smooth compact manifold. Then for some $d ...
- 1,438
8
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2
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771
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Separating trefoil knot on torus
Is it possible to draw a trefoil knot on the standard torus in ${\bf R}^3$ such that it separates the torus into two connected orientable components? And if not, why?
I can draw such a knot on a genus ...
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0
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0
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117
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Can one essentially cover a surface with two banners sewn together?
Let $S$ be an closed oriented surface (e.g. the sphere $S^2$). A banner with the pole $x_0\in S$ is an embedding $\beta: [0,1]\times [0,1]\to S\times [0,1]$ of the square into the thickening of the ...
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187
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Is there a Compact Non-Smoothable $4$-manifold that Topologically Immerses in $\mathbb{R}^5$?
Suppose that $M$ is a compact, connected topological $4$-manifold that has a topological immersion, i.e. local topological embedding, into $\mathbb{R}^5$. Then is $M$ necessarily smoothable? Note ...
- 559
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126
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Topological Isotopy Extension Theorem for Hypersurfaces
I seem to have worked out the details for the general topological isotopy extension theorem for tame $\mathbb{S}^n \subset \mathbb{R}^{n+1}$. Unfortunately it relies on most every bit of the smooth ...
- 559
7
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0
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81
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Slice vs ribbon genus for $g>0$ [duplicate]
Given a 1-knot $K$ the smooth slice genus is least integer $g$ so that $K$ is the boundary of a smooth genus $g$ surface in $S^3\times [0,\infty)$. The ribbon genus is defined in the same manner with ...
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314
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Theory of ends clarification
I am not an expert and so have decided to ask this question here. I have encountered 'ends'. It seems there are 2 such notions.
(1). à la Siebenmann and Freudenthal: An end $\epsilon$ of a non-compact ...
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6
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1
answer
298
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Beardon's version of Poincaré's theorem for fundamental polygons
I am currently trying to understand Beardon's proof for Poincaré's Theorem, which can be found in his book The Geometry of Discrete Groups. The last condition in the theorem is giving me a headache to ...
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6
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1
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358
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Gluing boundaries of 3-manifolds to get hyperbolic 3-manifold
Suppose $M$ is a compact, connected, orientable, aspherical 3-manifold, whose boundary $\partial M=S_1\cup S_2$ is a disjoint union of two surfaces $S_1,S_2$ with the same genus $g>1$. Denote by $...
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1
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253
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Connected sum with the fixed knot: injectivity of action
I would like to ask whether the connected sum operation with the fixed knot acts injectively on the isotopy classes of knots in the 3-sphere?
More precisely, given three knots $K_1$, $K_2$ and $K_3$ ...
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0
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88
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Realizing a Laurent polynomial as a dynamical zeta function or characteristic polynomial from an isolated surface singularity
Given a Laurent polynomial $\Delta(t) \in \mathbb{Z}[t, t^{-1}]$ satisfying:
$\Delta(1) = \pm 1$,
$\Delta(t^{-1}) = t^{\pm k} \Delta(t)$ (symmetry up to a unit),
can $\Delta(t)$ be realized as a ...
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0
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1
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375
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A foliation version of the Seifert conjecture
Edit:
Acording to comment of ThiKu I revise the question as follows:
Seifert conjecture asserts that every smooth or analytic vector field on $S^3$ possesses a closed orbit.
In this question I am ...
- 330
4
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0
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73
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Regularity of pseudo-Anosov flows near singular orbits
Let $f: \Sigma \to \Sigma$ be a pseudo-Anosov homeomorphism of a closed surface $\Sigma$. Let $V$ be the vector field on the mapping torus $M_f$ generating the suspension flow. I understand that there ...
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6
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4
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636
views
Heegaard splitting of 3-torus
I am trying to find Heegaard splitting of $3$-torus $T^3= S^1 \times S^1 \times S^1 $ with Heegaard genus $3$.
Thinking $S^1 \times S^1 \times S^1 $ as a cube identified the opposite faces , I can ...
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207
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Equivariant mapping class groups of the torus
I'd like to know the $G$-equivariant mapping class groups of the torus --- by which I mean the groups of connected components of the groups of $G$-equivariant diffeomorphisms,
$$
\pi_0\big( \mathrm{...
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5
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0
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206
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Understanding a theorem about tangential normal invariants of self homotopy equivalences of parallelizable manifolds
In the paper "On s-cobordisms of metacyclic prism manifolds" by S. Kwasik and R. Schultz, I am having problem understanding proposition 4.1. I am including the next paragraph also for better ...
5
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0
answers
211
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Massey-Cohen Immersion conjecture and its topological "corollary"
I was going through Cohen's proof of Immersion conjecture.
Cohen proves that any smooth $n$-manifold can be immersed in $\mathbb{R}^{2n-\alpha(n)}$, and this indeed is the tightest bound on the ...
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3
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1
answer
235
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Wall finiteness obstruction and components of Whitehead space
Given a CW complex $X$ that is a retract of a finite CW complex, the Wall finiteness obstruction is an element $w(X) \in \tilde{K}_0(\mathbb{Z}[\pi_1(X)])$ which vanishes if and only if $X$ is ...
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5
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1
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313
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Exact boundary of the blobs in Fig 9.5 of Indra's pearls
This question is about Figure 9.5 in Indra's Pearls.
In the figure, four blobs $a, A, b, B$ are drawn, along with their boundary curves.
My question is about the boundary curves of these blobs.
...
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3
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1
answer
396
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Evaluate $A p_1$ with Pontryagin Class on a 5-manifold
For the first Pontryagin Class $p_1$ that can be evaluated on a closed 4-manifold, are these true:
Lemma 1. $$\int_{M^5} A p_1 = 0 \mod 3,$$ when $A$ is $\mathbb{Z}/3$ valued 1-cochain, for a closed ...
- 10.8k
1
vote
0
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106
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Homeomorphisms between sphere arrangements arising from oriented matroids
Let $S^n \subset \mathbb R^{n+1}$ be the standard unit sphere, and define for each coordinate $i$ the "coordinate equator":
$$ S_i = S^n \cap \{x_i=0\} $$
Given a linear subspace $L \subset \...
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1
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0
answers
65
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Equation and curvature of a tape-like 3D manifold from UMAP embedding [closed]
I am currently working on a prime number classification method using 7 mathematical features. After dimensionality reduction with UMAP (into 3D), I observed that the prime numbers consistently appear ...
11
votes
1
answer
945
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unknotting number for k11a266
The unknotting number for k11a266 on the Hoste-Thistlethwaite table is given as 1. I simply do not believe this. I would be skeptical of 2. But I know this can be subtle. Does anyone have a ...
- 1,777
6
votes
1
answer
186
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The intersection number of hyperbolic metrics as geodesic currents
Let $\Gamma$ be a Fuchsian group such that $\mathbb H^2/\Gamma$ is topologically a closed surface $S$. Bonahon notably introduced the space of geodesic currents $C(\Gamma)$ as the space of $\Gamma$-...
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1
vote
1
answer
113
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Which $n$-stick unknots necessarily have a zero crossing projection?
Let $K$ be a particular embedding of a knot into $\mathbb{R}^3$, and let $[K]$ denote all knots equivalent (ambient isotopic) to $K$. The crossing number of $K$ is defined as $$c(K):=\min_{v\in S^2}~(\...
14
votes
1
answer
817
views
How complicated can the first integral homology of planar domains be?
Let $U \subseteq \mathbb{C}$ be a domain (i.e. connected open subset). I would like to know what is unconditionally known about $H_1(U,\mathbb{Z})$ (e.g. free, torsion-free, etc.?). In particular, I ...
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9
votes
1
answer
300
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Grassmannians, Schubert symbols and maps on cohomology
Let $n\in \mathbb{N}_0$.We consider the Grassmannian $G_n(\mathbb{C}^\infty)$. The cohomology groups of $G_n(\mathbb{C}^\infty)$ have a basis $\sigma_\lambda$ that is indexed by partitions $\lambda$ ...
- 3,077