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 ...

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2
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74 views

The Klee Trick for subsets of $\mathbb{R}^3$

I asked the question Is dimension given by the Klee trick ever sharp? That question remains unanswered, so I thought I might ask a slightly more concrete question along those lines. Given a metric ...
5
votes
2answers
206 views

Using Discrete Morse Theory to represent hom classes

In "vanilla" Morse theory, you can construct cycles representing integral homology classes of a smooth manifold from a Morse function on the manifold (by looking at the flow into/out of the critical ...
1
vote
0answers
44 views

square tiled surfaces: Counting Saddle Connections vs Counting Square-Tiled Surfaces

I am reading Prime arithmetic Teichmuller discs in H(2) where they discuss the Siegel-Veech constant in a stratum $\mathcal{H}(2)$ of surfaces. However I see two definitions of Siegel-Veech constant: ...
1
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0answers
49 views

Decomposition of Lefschetz fibrations into surface bundles and Lefschetz fibrations over spheres

Assume that $M$ is a Lefschetz fibration with fiber genus $g$ over a base of genus $h>0$ (with at least one singular fiber). What obstructions exist to decomposing $M$ into a surface bundle $S$ of ...
6
votes
1answer
223 views

Software for computing Thurston's unit ball

Is there any software which can be used for computing Thurston's unit ball(for second homology of 3-manifolds) of link complements? In particular can I do that with SnapPy? PS: even a table for ...
3
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0answers
57 views

Real laminations on a 4-punctured sphere

Fix a triangulation $T$ of the 4-punctured sphere. (Formally, an ideal triangulation, but taking a combinatorial viewpoint, we may as well just fix a triangulation of the sphere with 4 vertices and ...
4
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1answer
166 views

Ends of Coxeter Groups

It is known after Stallings that a group can have 0, 1, 2 or infinitely many ends. Are there known results on the space of ends of a Coxeter group?
2
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0answers
43 views

Definition of the dual spider number and the formula for the first chern class of the triangle

In the process of trying to understand various maps in Heegaard Floer homology I got stuck on the definition of the dual spider number, which, it seems to me, has a combinatorial definition directly ...
5
votes
1answer
225 views

Proving that the Jones polynomial is q-holonomic

The Jones polynomial is known to have many different interpretations or definitions, by now. There are connections with QFT, quantum groups, Hilbert schemes, Cherednik algebras, etc. My question is ...
2
votes
1answer
63 views

Quandle colorings under Reidemeister moves

Let $D$ be a knot diagram and $Q$ a quandle. We use $c$ to denote a fixed coloring of $D$ with $Q$. If $D'$ is another knot diagram of the same knot, and $R_1$ is a sequence of Reidemeister moves ...
2
votes
2answers
134 views

Quasi-isometry and left invariant orderability for groups

Is the property of left invariant orderability for finitely generated groups preserved by quasi-isometrics? More precisely, if $G$ is a left orderable (finitely generated) group and $H$ is a ...
18
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0answers
215 views

Topological description of inverting a knot

The first figure shows an offset overhand knot. To tie it, you simply bring the two cords together and make an overhand knot in them. It's more secure than it looks, and several climbers have been ...
3
votes
5answers
235 views

Existence of orientation preserving, finite order self homeomorphism on a genus 2 surface without fixed point

Let $M$ be a compact 2-manifold of genus 2. Does there exist an orientation preserving homeomorphism $f:M\to M$, so that $f^n=id$ for some integer $n$, and $f$ doesn't have fixed points? Using ...
1
vote
0answers
71 views

Disks in Flat Embeddings of Graphs in $\mathbb{R}^3$

Robertson, Seymour and Thomas proved that any linkless graph $G$ has a flat embedding in $\mathbb{R}^3$ (see for example A survey of linkless embeddings). An embedding of $G$ is flat if for any cycle ...
5
votes
1answer
169 views

(Smooth) Borel Conjecture for 4-dimensional torus

Given an aspherical 4-dimensional closed manifold $M$ with fundamental group $\mathbb{Z}^4$, it is homotopy-equivalent to $T^4 = S^1 \times \ldots S^1$, the 4-dimensional torus. Question 1: Since I ...
4
votes
2answers
92 views

Which criteria guarantee an orthogonal circuit in $\mathbb R^3$ to be rigid?

For $n\ge4$, define an orthogonal circuit or O-circuit as a closed circuit of $n$ unit segments in $\mathbb R^3$ such that any two neighboring segments form a right angle. (Physically this could be ...
3
votes
0answers
57 views

Different definitions of linkless graphs

Robertson, Seymour and Thomas defined linkless embeddings of graphs as follows: An embedding of $G$ is linkless if every pair of disjoint circuits of $G$ have zero linking number (see here). However ...
1
vote
1answer
87 views

Difference between straight and piecewise linear and continuous embeddings of graphs / complexes in d-dimensional space?

Here is my main question: what is the difference between "straight" and "piecewise linear" and "continuous" embeddings of graphs/complexes in d-dimensional space? Moreover I would like to know if any ...
12
votes
1answer
231 views

Is a generic closed orientable hyperbolic 3-manifold Haken?

My question is as follows: "Is a generic closed orientable hyperbolic 3-manifold Haken?" Of course the word 'generic' can be interpreted in many ways, and the answer might depend on the way how one ...
8
votes
1answer
209 views

Second homology of mapping class group of genus 3

In a survey paper of Korkmaz it is stated that $H_2(\mathrm{Mod}_3)$ is either $\Bbb Z$ or $\Bbb Z \oplus \Bbb Z_2$, but I was not able to find out a precise computation of this group (resolving the ...
2
votes
0answers
38 views

Any results on rayless simplicial complexes?

We define a closed ray in a topological space $X$ to be its closed subset homeomorphic to the real half-line $[0,\infty)\subseteq \mathbb{R}$. Call a topological space $X$ rayless if it does not ...
2
votes
1answer
86 views

Criteria for abstract polytopes to be convex polytope

Suppose I have an abstract polytope defined by a poset. Are there any methods for determining whether the abstract polytope can be geometrically realized as convex-hull on its set of vertices?
0
votes
0answers
64 views

Condensation points of orbits of roots of unity

For a fixed $n\in \mathbb{N}$ we consider the set of $n$-roots of unity $R(n)=\{z\in S^1; z^n=1\}$. It splits into mutually disjoint orbits. Let $R=\bigcup_{n=0}^{\infty} R(2^n-1)$. For each orbit in ...
2
votes
0answers
158 views

Bott's Formula for Grassmannians

Bott's Formula gives the dimension of the cohomology $H^{q}(\mathbb{P}^{n}, \Omega_{\mathbb{P}^{n}}^{p}(k))$ of the $k$-twisted sheaf of $p$-differential forms on the projective space ...
15
votes
2answers
292 views

Classification of tangles?

Has anybody done any work on making a classification of low-complexity tangles, analogous to the work for knots and links? I expect most of the small ones to be rational, and those that aren't ...
1
vote
0answers
73 views

Singular leaf of Strebel differential

Let $R$ be a Riemann surface.Let $\gamma$ be a loop which is non-trivial in $H_{1}(R,\mathbb{Z})$. By the Jenkins–Strebel Theorem we know the following: there exists a holomorphic quadratic ...
9
votes
2answers
248 views

Heegaard genera of arithmetic 3-manifolds

UPDATE: Because I was hoping that state the question as concisely as possible, the original post did not include a precise definition of arithmetic 3-manifold only a reference to Maclachlan and ...
2
votes
1answer
64 views

Characterization of the medial axis of a surface

I would like to know if the following "characterization" of the medial axis of a surface is correct, and if so, how to prove it. Let $S$ be a continuous, piecewise smooth, compact surface embedded in ...
3
votes
2answers
226 views

Pseudo-manifolds and homology

Is there a good reference for the proof that the cobordism group of pseudo-manifolds is isomorphic to the singular homology group? I was looking for a more geometrical definition of homology and ...
12
votes
0answers
154 views

What do tangles teach us about braids?

A braid is a smooth level-preserving embedding $f\colon\, \{1,2,\ldots,n\}\times[0,1]\hookrightarrow \mathbb{R}^2 \times [0,1]$ such that $f(k,0)=(k,0)$ and $f(k,1) \in \{1,2,\ldots,n\} \times ...
2
votes
1answer
183 views

Map from homotopy sphere with lifting property induces surjections on homotopy groups. Is it weak equivalence?

Let $E$ be homotopy equivalent to a $k$-sphere. Let $q\colon E\to X$ be a map such that given any continuous $f\colon C\to X$ from a compact space $C$, there exists (a non-unique) $\tilde{f}\colon ...
3
votes
1answer
139 views

laminations and branched surfaces

I am looking for a reference for this question: given a branched surface in a 3-manifold, how we can construct a lamination fully carried by that branched surface. any comments would be appreciated. ...
5
votes
0answers
216 views

Quotienting disk inside sphere result in sphere

Let $S^k$ be a topological $k$-dimensional sphere. Let $D^k$ be a $k$ dimensional disk that includes in $S^k$. Let $q: D^k \to D^r$ be a map and $r \leq k$. Let $$W = S^k \sqcup D^r/\sim$$ where ...
2
votes
1answer
81 views

Going Back-and-Forth Between Different Expressions/“Representations” for Open Books.

I am trying to have a better understanding of how one goes , "travels" between the different formats/layouts of open books for a fixed given 3-manifold M; between the abstract type and the "actual" ...
4
votes
1answer
131 views

Most Regularity of a Polygon

Conseider $n$ electrons in an empty sphere. What structure do they make? This question have two cases: (i) if electrons should be sit on the boundry of sphere (one can suppose that the boundry of ...
11
votes
1answer
152 views

Max flow, min cut on manifolds

If a graph has some half edges marked "input" and some half edges marked "output", it is well known that the smallest number of edges which must be cut to disconnect input from output is equal to the ...
5
votes
1answer
96 views

In the definition of the Heegard Floer surgery exact triangle, what exactly is the correspondence between Whitney triangles and periodic domains?

I'm reading Osváth-Szabó's notes on Heegard Floer homology, in particular about the surgery exact triangle. On page 14 (numbered 42 on the document), they describe an isomorphism between the space of ...
3
votes
2answers
205 views

Is the hypersurface satisfying $\langle x-x_0,\nu\rangle>0$ diffeomorphic to sphere?

Let $p:M\to \mathbb{R}^{n+1}$ be the closed immersed hypersurface. Is the following thing right? If there exists a point $x_0$ in $\mathbb{R}^{n+1}$ such that $\langle x(p)-x_0,\nu(p)\rangle>0$ ...
12
votes
1answer
319 views

Why are Witten-Reshetikhin-Turaev invariants expected to be integral?

A Witten-Reshetikhin-Turaev (WRT) Invariant $\tau_{M,L}^G(\xi)\in\mathbb{C}$ is an invariant of closed oriented 3-manifold $M$ containing a framed link $L$, where $G$ is a simple Lie group, and $\xi$ ...
4
votes
2answers
243 views

A question about Dehn surgery and Brieskorn homology 3-spheres

I have been learning about Brieskorn homology 3-spheres $\Sigma(a_1,...,a_n)$ and Seifert manifolds. My reference is the first few pages of Saveliev's "Invariants of Homology 3-spheres." If I ...
4
votes
2answers
77 views

Is it known which links have Seifert fibered complements?

I believe many such links can be constructed by looking at a foliation similar to the hopf fibration, but the wrapping leaves replaced with $(p,q)$ torus knots. However, I'm interested in particular ...
5
votes
2answers
213 views

What 3-manifolds can be obtained by gluing $ S^1 \times P $ and two copies of $S^1 \times D^2$

Let P denote the pair of pants e.g. a sphere minus three small discs $D_1$,$D_2$,$D_3$ about marked points $x_1,x_2,x_3$. I then consider $P \times S^1$. We have boundary components $T_1$,$T_2$,$T_3$. ...
2
votes
1answer
183 views

Simple connectedness of $\mathbb{C}P^2$ intersected with an affine subspace

The complex projective plane $\mathbb{C}P^2$ can be thought of as the set of rank one 3-by-3 Hermitian matrices with norm one, i.e., $\mathbb{C}P^2 = \{xx^* : x \in \mathbb{C}^3, x^*x=1 \}$. As such, ...
17
votes
1answer
577 views

Topological transversality

Warmup question: Let us say that two continuous functions $f,g:[0,1]\to \mathbb R$ are topologically transverse if their difference $f-g$ has only finitely many zeros, and each zero separates an ...
3
votes
0answers
211 views

Does the following object has a name in algebraic geometry?

Suppose $X$ is a projective variety and $D$ is a smooth divisor and let $L = \mathcal{O}(D)$ be the line bundle corresponding to $D$. Consider $X \times \mathbb{P}^1$ with the line bundle ...
1
vote
1answer
121 views

Ratner theorem and dense geodesic planes in hyperbolic manifolds

Suppose we have a closed hyperbolic $3$-manifold $M$. For any $x\in M$ and plane $\pi$ in $T_xM$ we consider $P$ the geodesic plane exp$(\pi)$ originating from $\pi$. For any $p\in \pi$ we consider ...
4
votes
1answer
596 views

solvable word problem without algorithm

Let $G$ be a finitely generated group. I wonder if there are examples where: 1) The word problem is known to be solvable in $G$ but there is no algorithm known. 2) The word problem is known to be ...
3
votes
1answer
193 views

Knot invariants in 3-manifolds that are not $\mathbb{R}^3$ or $S^3$ or $B^3$?

This is just a reference request; I have no sharp mathematical question. Inspired by the $(3+)$-year old MO question, In knot theory: Benefits of working in $S^3$ instead of $\mathbb{R}^3$?, I would ...
2
votes
0answers
83 views

Pure braid groups of the complement of a lattice in the complex plane: generators and relations

Where can I find a presentation (by `natural' generators and relations between them) of the pure braid groups $PB_n(S)$ (for $n>0$) of $S=\mathbb C\setminus (\mathbb Z\oplus i \mathbb Z)$? Thanks ...
12
votes
0answers
336 views

What is the determinant of Poincare duality?

For a complex $C^{\bullet}$ of finite dimensional vector spaces, one has a determinant $$|C^\bullet|:= \bigotimes \left(\Lambda^{top} C^i\right)^{(-1)^i}$$ functorial with respect to ...