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).
4,154
questions
2
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Corollary in Rasmussen's paper about $s$-grading of Lee's canonical generators
In Jacob Rasmussen's paper Khovanov homology and the slice genus, he states as Corollary 3.6 that $s(\mathfrak s_o)=s(\mathfrak s_{\bar o})=s_{min}(K)$, where $s$ is the $s$-grading and $\mathfrak s_o,...
- 213
4
votes
1
answer
155
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Existence of tubular neighborhood of singular complex subvariety
Let $X$ be a smooth complex projective manifold and let $Y$ be a closed subvariety of $X$ with $y\in Y$ a fixed point. Does there exist an open neighborhood $U$ of $Y$ such that $\pi_1(Y,y)\to \pi_1(U,...
- 507
3
votes
1
answer
220
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Handle attachment information from Morse function and triangulation
First, allow me to setup the relevant information. It is well known that a Morse function $f:M\to\mathbb{R}$ induces a handle decomposition of $M$.
For simplicity, let's restrict for now to the ...
- 159
7
votes
1
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178
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Calculating the Seifert framing for an exceptional fiber in a Seifert-fibered integer homology 3-sphere
Let $Y=\Sigma(\alpha_{1},\dots,\alpha_{n})$ be a Seifert fibered homology 3-sphere, let $\pi:Y\to\Sigma$ denote the projection to the orbifold surface, and let $T\subset Y$ be the pre-image under $\pi$...
- 299
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1
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228
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Bounds for the crossing number in terms of the braid index?
Is there a lower bound on the crossing number of a knot (resp., link) with braid index $b$?
For knots, I believe the smallest crossing number for braid index 2 is 3, the smallest crossing number for ...
- 8,994
3
votes
1
answer
187
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Maps of surfaces to CAT(0) cube complexes
Let $C$ be a locally CAT(0) cube complex. Let $f\colon \Sigma_g \rightarrow C$ be a continuous map from a closed oriented genus $g \geq 1$ surface to $C$. Is it always possible to find a cube ...
- 31
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1
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309
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Simplified Bing's house
Bing's house is an example of contractible 2-complex in $\mathbb{R}^3$.
One may think that it is a surface without boundary that has two types of singularities: tripod curves — curves where three ...
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10
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0
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201
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Bi-Lipschitz mappings
Assume that we have a bi-Lipschitz mapping $f:\bar{\mathbb{B}}^n(0,1)\to\mathbb{R}^n$. The mapping need not be smooth anywhere and it may happen that it cannot be extended to a homeomorphism of a ...
- 27.3k
3
votes
1
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346
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Boundaries of subsets of simply-connected domains
I am trying to prove the following assertion: Let $B$ be a simply-connected set, and let $B' \subsetneq B$ be a proper open connected subset. Then, there exists a point $x \in \partial B'$ of the ...
3
votes
0
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68
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Discreteness of volumes of boundary-parabolic representations
Suppose $M$ is a cusped hyperbolic $3$-manifold of finite volume. Let $\mathfrak{R}_0(M)$ be the space of boundary-parabolic representations $\rho : \pi_1(M) \to \operatorname{PSL}_2(\mathbb C)$. Is ...
- 2,483
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228
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The second coefficient of the Conway polynomial from Knot Floer homology
Let $\nabla_K(z)$ be the Conway polynomial and $\Delta_K(t)$ be the Alexander polynomial normalized by $\Delta_K(t)=\Delta_K(t^{-1})$ and $\Delta_K(1)=1$,
These invariants are equivalent and they are ...
- 341
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2
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254
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Surjections from genus $n$ surface group to free group of rank $n$
Let $\Sigma_n$ be a genus $n$ surface, let $\mathcal{H}_n$ be a genus $n$ handle body, and let $F_n$ be a free group of rank $n$. Fix an identification of $\pi_1(\mathcal{H}_n)$ with $F_n$. I know ...
- 73
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1
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Definition of Thurston's skinning map
A key construction in Thurston's proof of the existence of hyperbolic structures on Haken manifolds is the so-called "skinning map" associated to a 3-manifold $M$ with boundary whose ...
- 155
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1
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"Neck cutting" and why gauge theory doesn't work on homotopy 4-spheres
I attended a talk recently and the speaker said, essentially, that gauge theory invariants are expected to never be able to detect exotic 4-spheres because they always vanish, for a reason related to ...
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2
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Generate $\mathrm{Mod}(S_g)$ by two Dehn twists
Let $S_g$ be a closed orientable surface of genus $g>1$.
How can one prove that its mapping class group $\mathrm{Mod}(S_g)$
is not generated by two Dehn twists?
A pair of simple closed curves in $...
- 965
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0
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86
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Terminology for discrete subgroups of PSL(2,k), where k is a non-archimedean local field
$\DeclareMathOperator\PSL{PSL}$I'm asking about terminology for discrete subgroups of $\PSL(2,k)$, where $k$ is a non-archimedean local field.
As it is rather clumsy to have to use such expressions ...
- 586
1
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0
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107
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Instantons on the 4-sphere with respect to other Riemannian metrics
It is known that the moduli space of $\text{SU}(2)$ instantons of charge $1$ on $S^4$ is diffeomorphic to the five-ball $B^5$ if $S^4$ is endowed with the round metric.
Question: what does the moduli ...
- 975
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Can every homeomorphism of the 4-ball fixing the boundary be approximated by diffeomorphisms smoothly isotopic to the identity?
Let $B^{n}$ denote the euclidean closed ball of dimension $n$. Alexander's trick shows that every homeomorphism of $B^{n}$ fixing $\partial B^n$ is $C^{0}$-isotopic to the identity via a boundary ...
- 141
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Is there any functoriality of Stallings' twists?
Suppose $L$ is an oriented link and $L'$ is a link obtained from the Stallings' twists.
Are there any functoriality of the twist operation on the links, such as functoriality between (1) homology ...
- 1
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106
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Finite homology of a homogeneous space
Let $\Gamma$ be a cocompact lattice in $\operatorname{SL}(2,\mathbb R)$ and $X=\operatorname{SL}(2,\mathbb R)/\Gamma$ be the underlying homogeneous space. Can the homology group $H_1(X,\mathbb Z)$ be ...
5
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89
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Irreducible factors of the A-polynomial
The A-polynomial $A_K$ of a knot $K$ describes the irreducible "non-abelian" components of the $SL(2)$-character variety of $S^3-K.$
Does anyone know a knot K for which $A_K$ has more ...
- 2,370
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0
answers
295
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Fundamental group of blow-ups
Let $M$ be a simply-connected compact complex manifold of dimension three and $C$ is a smooth complex curve in $M$.
Let $M'$ be the blow-up of $M$ along $C$.
My question is:
Is $M'$ also simply-...
- 1,821
3
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1
answer
180
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Squier's conjecture on Burau at roots of unity
In Squier's short, yet influential, paper about the Burau representation, he made two conjectures that might have provided a proof for the faithfulness of the Burau representation (which we now know ...
- 1,267
3
votes
1
answer
213
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Example of a non $\pi_1$-injective, degree one, self-map of a three-manifold
All manifolds will be assumed to be closed, oriented, and connected.
Let $f\colon M\to M$ be a map of degree $\pm 1$. It is not hard to show that $\pi_1(f)$ is surjective.
What is an example of a non ...
- 1,087
4
votes
0
answers
259
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Does this "join-like complex" of $K_5$ and $K_3$ embed in $\Bbb R^4$?
Consider the following 2-dimensional CW-complex: its 1-skeleton is $K_8$, which we write as an edge-disjoint union $K_5\cup K_{5,3}\cup K_3$.
Then for any two edges $ab\in E(K_5)$ and $cd\in E(K_3)$ ...
- 12.6k
1
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2
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158
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Properly embedded annuli in genus two handlebody?
Is it possible to have a properly embedded annulus $A$ in a genus two handlebody $V$ such that the two boundary curves of $A$ in $\partial V$ represent different isotopy classes of curves on $\partial ...
- 379
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3
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Computation on characteristic classes
I am organizing a reading seminar on characteristic classes. The audience in the seminar is interested in symplectic and contact manifolds. I work in categorification and would like to compute some ...
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1
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142
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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 $(X,d)$. ...
- 322
14
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1
answer
839
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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 ...
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80
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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 ...
- 7,755
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0
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129
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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 $...
- 1,065
2
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0
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80
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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
226
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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 ...
- 12.6k
4
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2
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305
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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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1
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366
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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 $p_i$ ...
- 345
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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 ...
- 258
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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 ...
- 1,825
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1
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270
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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
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203
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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 ...
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4
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1
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401
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Faithful locally free circle actions on a torus must be free?
Is it true that every faithful and locally smooth action $S^1 \curvearrowright T^n$ is free?
I know such an action must induce an injection $\rho:\pi_1(S^1)\to\pi_1(T^n)$.
Another related question is: ...
- 375
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2
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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 ...
- 10.6k
2
votes
1
answer
66
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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 $V_d \subset X$ ...
- 6,337
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253
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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 ...
- 975
1
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1
answer
244
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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 ...
- 215
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2
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552
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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 ...
- 333
1
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1
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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/...
- 187
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1
answer
256
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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 $b_2(W)=1$? ($\cdots$...
- 475
2
votes
0
answers
237
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 ...
- 21
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1
answer
264
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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 ...
- 606
2
votes
0
answers
104
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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:
$$\...
- 5,201