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).
4,129
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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
votes
1
answer
318
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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?
3
votes
0
answers
181
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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 ...
8
votes
1
answer
665
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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 ...
3
votes
1
answer
159
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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 ...
3
votes
2
answers
417
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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 torsion-...
20
votes
0
answers
493
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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 ...
6
votes
5
answers
768
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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 ...
2
votes
0
answers
138
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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 $...
6
votes
1
answer
442
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(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 ...
6
votes
2
answers
164
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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
0
answers
174
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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
1
answer
418
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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
1
answer
499
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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
1
answer
374
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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
0
answers
68
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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 ...
3
votes
1
answer
269
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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
0
answers
89
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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 $...
5
votes
1
answer
953
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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 $\mathbb{P}_{\...
18
votes
3
answers
623
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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 ...
3
votes
1
answer
204
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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 ...
10
votes
2
answers
634
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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
1
answer
113
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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
2
answers
606
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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 ...
18
votes
0
answers
460
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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 $f(k,1) \in \{1,2,\ldots,n\} \times \{1\}$....
2
votes
1
answer
223
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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 C\...
14
votes
1
answer
438
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Bloch group, hyperbolic manifolds and rigidity
I have some questions concerning the hyperbolic geometry side of the rigidity question for $K_3$ which asks if the natural map $K_3^{\operatorname{ind}}(\overline{\mathbb{Q}})\to K_3^{\operatorname{...
4
votes
1
answer
322
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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
0
answers
257
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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 $S^...
2
votes
1
answer
143
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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
1
answer
271
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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 ...
12
votes
1
answer
363
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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 ...
7
votes
1
answer
436
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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
2
answers
239
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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 $\langle x(p)-x_0,\nu(p)\rangle>0$ ...
15
votes
1
answer
1k
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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 $\xi$ ...
6
votes
2
answers
996
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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 ...
6
votes
2
answers
393
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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
2
answers
491
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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
1
answer
195
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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, ...
18
votes
1
answer
1k
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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
0
answers
227
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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 $\mathcal{O}(...
1
vote
1
answer
264
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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 ...
5
votes
1
answer
874
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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 ...
5
votes
1
answer
456
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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
0
answers
109
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 ...
15
votes
0
answers
586
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What is the determinant of Poincaré 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 quasi-...
10
votes
0
answers
1k
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Some questions about geodesic lamination
I'm learning geodesic laminations on surfaces. Here are some questions I thought a lot but could not understand well.
We consider a complete finite area hyperbolic surface $S$ w/o geodesic boundary. ...
32
votes
2
answers
2k
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The Jones polynomial at specific values of $t$
I've been calculating some Jones polynomials lately and I was just curious if there was a "physical" (or, rather, geometric) meaning to evaluating the Jones polynomial at a particular value of $t$.
...
4
votes
1
answer
593
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Manifolds such that every homeomorphism of a submanifold to itself extends to the full manifold
Let manifold $S$ (connected, without boundary) have next property: for every submanifold $D \subset S$ (connected, compact, without boundary), every homeomorphism $f:D \to D$ extends to a ...
13
votes
1
answer
1k
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Representation varieties of 3-manifold groups in $\mathrm{SL}(n,\mathbb{C})$
I am looking at the variety of representations of the fundamental group of a hyperbolic 3-manifold into $\mathrm{SL}(n,\mathbb{C})$:
$$\mathrm{Hom}(\pi_1(M), \mathrm{SL}(n,{\mathbb C}))$$
It is known ...