**6**

votes

**0**answers

178 views

### Right-angled polytopes

%This question is motivated by the little discussion here at the bottom.
The following thing are known about hyperbolic right-angled polytopes:
Compact hyperbolic right-angled polytopes do not ...

**4**

votes

**0**answers

98 views

### Is the normalised Kauffman bracket more powerful than the Kauffman bracket?

The Kauffman bracket polynomial for a knot diagram $D$ is a Laurent polynomial $\langle D \rangle \in \mathbb{Z}[A, A^{-1}]$. Although it is invariant under Reidemeister moves of type II and III, ...

**7**

votes

**2**answers

162 views

### How does Thurston's Orbifold Geometrization imply that knots with meridional rank 2 are 2-bridge?

Problem 1.11 of Kirby's list asks whether every knot that has a knot group
which can be generated by n meridians, but not less than n, is an n-bridge
knot. There is a one-sentence update, saying that ...

**1**

vote

**0**answers

356 views

### What is a good algorithm to measure similarity between two dynamic graphs?

I am using graphs to represent structure present in a scene. The vertices represent the objects in the scene and the edges represent the relationship between two nodes(touching, overlapping, none). ...

**2**

votes

**0**answers

156 views

### Stratification of a smooth map

So, this is an exercise. But from math.stackexchange I have been suggested to post this question here.
To find the Thom-Boardman stratification of the smooth map
...

**5**

votes

**2**answers

263 views

### Physical meaning of the integral cohomology condition in Souriau-Kostant pre-quantization?

The question is in the title. The form of the condition looks like the Bohr-Sommerfeld quantization formula of angular momentum, is there a link between the two formulas?

**1**

vote

**1**answer

128 views

### Two links with the same signatures but unknown if they are related by Kirby moves

I am wondering if there are links $L_1, L_2$ in the sphere $S^3$ such that:
the signatures of $L_1, L_2$ are known.
we do not know if they are related by Kirby moves.
If so, could you specify the ...

**2**

votes

**1**answer

244 views

### Complex continued fractions with given digits

For which sets of Gaussian integers $A\subseteq\mathbb{Z}[i]$ is the set of continued fractions with digits in $A$
$$ C(A)=\{a_0+\cfrac{1}{a_1+\cfrac{1}{a_2+\cdots}}\mid a_i\in A\} $$
totally ...

**14**

votes

**1**answer

586 views

### A concrete realization of the nontrivial 2-sphere bundle over the 5-sphere?

Since $\pi_4 (PU(2)) = \pi_4 (SO(3)) = {\mathbb Z}_2$, the two-element group,
we know that half of the two-sphere bundles over the 5-sphere $S^5$ are trivial
and the other half are non-trivial and ...

**3**

votes

**1**answer

248 views

### Geometrization & JSJ decomposition with boundary

Is there any paper where I can find a good explanation of the JSJ decomposition, the geometrization theorem and the relations between them when the manifold has nonempty (and non necessarily toroidal) ...

**4**

votes

**1**answer

234 views

### Casson invariant and signature

In W. Neumann, J. Wahl, "Casson invariant of links of singularities",
Comment. Math. Helv.,1990, Vol. 65, Issue 1, pp 58-78 some connection between the Casson invariant and the signature is ...

**0**

votes

**1**answer

162 views

### A question on Cayley graphs and hyperbolic 3-manifolds

There are two hyperbolic closed 3-manifolds, but I don't know whether they are homeomorphic or not. The only thing I know is that the Cayley graphs of their fundamental groups are quasi-isometric.
...

**1**

vote

**0**answers

52 views

### Asymptotic translation length on the complex of curves and the forgetful map

Let $\phi$ be a pseudo-Anosov on a surface with punctures with asymptotic translation length $L(\phi)$. If under the forgetful map $\phi$ restricts to a pseudo-Anosov $\hat{\phi}$ on a connected ...

**0**

votes

**1**answer

195 views

### question about the developing map

I'm having some trouble finding literature on the developing map.
All the sources I could find on it seem to refer to thurston's definition in either:
http://www.ucl.ac.uk/~ucahhjr/Notes/Essay.pdf
or
...

**7**

votes

**3**answers

226 views

### A version of Lusternik–Schnirelmann category for good open covers

Recall that the Lusternik–Schnirelmann category (or LS-category) of a space is the integer $n$ such that there is an open cover by $n+1$ open sets which have nullhomotopic inclusions, and no such ...

**5**

votes

**1**answer

171 views

### Voronoi cells and the dual complexes in Riemannian manifolds

I would like to use some "intuitively clear" properties of Voronoi cells in general Riemannian manifolds, but I have trouble finding references.
Let $(X,d)$ be a connected Riemannian manifold and ...

**4**

votes

**0**answers

194 views

### Is there an “exponential law” for differentiable maps between smooth manifolds?

Although it seems like a textbook question, I was not able to find a textbook or even a research article answering the following question:
Let $M$, $N$ and $P$ be finite-dimensional smooth manifolds ...

**4**

votes

**0**answers

129 views

### Haken manifolds and characterising sutured manifold hierarchies

In Gabai's paper (Knot Theory and Manifolds Lecture Notes in Mathematics Volume 1144, 1985, pp 14-17 An internal hierarchy for 3-manifolds) he considers sutured manifold decompositions of Haken ...

**4**

votes

**1**answer

211 views

### space of simple loops in the plane

Consider the space $S$ of smooth free simple (non-self-intersecting) loops in the plane $\mathbb{R}^2.$ By Grayson's theorem, $S$ is connected, but is more known about its topology? Is it known to be ...

**3**

votes

**1**answer

235 views

### Iterated Lefschetz numbers

Given a pseudo Anosov mapping class $f:S_{g,n}\rightarrow S_{g,n}$ is the Lefschetz number for $f^m$ negative for some $m$ depending only on $(g,n)$?
The Lefschetz number of a mapping class $f$ can ...

**8**

votes

**1**answer

140 views

### Counterexamples to analogue of Cannon conjecture in higher dimensions

It is known that a group $G$ acts geometrically on $\mathbb{H}^2$ if and only if $G$ is word-hyperbolic and its boundary $\partial G$ is homeomorphic to $S^1$.
The analogous statement for ...

**6**

votes

**3**answers

299 views

### Are non-isomorphic covers of riemann surfaces also generally nonisomorphic as riemann surfaces?

Suppose you've got a Riemann surface $E$, and two topological covers $X,Y\rightarrow E$. Suppose $X,Y$ are nonisomorphic topological covers of $E$, then would you expect $X,Y$ as Riemann surfaces ...

**4**

votes

**1**answer

137 views

### Lipschitz Approximation to a PW Smooth Map

Suppose I have a triangulated smooth manifold, $\tau : |K| \rightarrow M$ (so that $\tau | _{\sigma}$ is smooth for each $\sigma \in K$), and a piecewise smooth map, $f: M \rightarrow \mathbb{R}^n$. ...

**0**

votes

**1**answer

92 views

### taut foliations and the existence of total transversals

A codimension one foliation $\cal F$ on a smooth manifold $M$ is taut if every leaf of $\cal F$ meets a closed transversal (i.e., a simple closed curve that is everywhere transversal to the leaves of ...

**4**

votes

**0**answers

202 views

### Relationship between virtual cohomological dimension and tautological rings for moduli spaces of curves

Here's the short version of the question. For $M_{g,n}$, $M_{g,n}^{rt}$, $M_{g,n}^{ct}$ and $\overline M_{g,n}$ it seems that the virtual cohomological dimension is given by the complex dimension plus ...

**5**

votes

**1**answer

101 views

### Minimum dilatation pseudo-anosovs on non-orientable surfaces

Is anything known about minimum dilatation pseudo-anosovs on non-orientable surfaces?
More specifically it is known whether the asymptotic behavior for log(minimal dilatation) is 1/genus? (The lower ...

**2**

votes

**1**answer

157 views

### Group actions on trees and translates under hyperbolic elements

I have the following question regarding group actions on trees to which I suspect the answer to be "yes", but it could very well be that extra conditions are required (it is certainly true for free ...

**3**

votes

**2**answers

117 views

### intersection of Whitney stratifications

Let $X$ be an oriented smooth manifold with dimension $n$. If $U$ and $V$ are two oriented closed submanifolds of $X$ and $U$ is transverse to $V$ in $X$. Then $U\cap V$ (suppose the intersection is ...

**2**

votes

**1**answer

151 views

### winding number for outer-pointing normal

While trying to characterize the complexity of a closed differentiable curve (for a path planning application), I've been using a notion which is similar in spirit to the winding number of a curve. ...

**0**

votes

**0**answers

73 views

### Open Book Decompositions of M^3's : Finding the Projection Map (Hope in Coordinates) in an Abstract Open Book

all:
I want to know how to find out , hopefully in coordinates, (but I'll take what's available) , the description of the projection map in an abstract open-book decomposition.
Open book ...

**5**

votes

**1**answer

210 views

### Andreev's Theorem and Thurston's hyperbolization theorem

I am attempting to get to grips with Thurston's hyperbolization theorem for Haken $3$--manifolds. In particular I was looking at the section related to gluing up along hierarchy surfaces in Otal, ...

**3**

votes

**1**answer

152 views

### Classification of fiber-preserving branched covers between Seifert fibered integer homology spheres

Is there an easy classification (and proof) of the possible branched covers between Seifert fibered integer homology spheres which are fiber-preserving and branched over fibers (or at least what the ...

**3**

votes

**0**answers

175 views

### Are there CW structures on homotopy limits of CW maps?

Consider the diagram of finite CW complexes $X \stackrel{f}\leftarrow Y$ where $f$ is a cellular map and note that its homotopy colimit is precisely the mapping cylinder
$$C_H = \frac{X \sqcup (Y ...

**6**

votes

**0**answers

112 views

### Torsion in Whitehead group

Let $\pi$ be a finite group of odd order. What do we know about the torsion subgroup of $Wh(\pi)$? I am especially interested in the $2$-primary part. Is it always trivial?

**4**

votes

**2**answers

193 views

### A question about trees and planar graphs

Let T be a tree which satisfies the following conditions. (A) The set of vertices of T is denumerably infinite. (B) Each vertex of T is an end-point of at most finitely many edges of T. Does there ...

**5**

votes

**1**answer

205 views

### Standard (special) spines and hyperbolic structure on 3-manifolds

My question relates to constructing angled triangulations or hyperbolic triangulations for $3$--manifolds. Briefly, an angle triangulation can be considered as an assignment of a real number (called ...

**12**

votes

**2**answers

785 views

### Converse of Poincaré-Hopf theorem

Let $M$ be a connected, compact, oriented manifold of dimension $n<7$. If any two maps $M \to M$ having equal degrees are homotopic, must $M$ be diffeomorphic to the $n$-sphere?

**8**

votes

**1**answer

311 views

### certain trigonometric homeomorphisms

Are there any simple characterizations of rational functions $f(x,y)$ with real coefficients such that $\theta\mapsto f(\cos\theta,\sin\theta)$ is a homeomorphism from $\mathbb R\bmod 2\pi$ to ...

**0**

votes

**0**answers

89 views

### Distance at least $n$ geodesics on surface

everyone. Let $S$ be a closed orientable 2-surface with genus at least 2. Let
$\{C_{i}\mid i=1,...,3g-3\}$ be a pants decomposition of $S$. Suppose that $\{l_{C_{i}},\tau_{C_{i}}, for~ ...

**2**

votes

**1**answer

195 views

### Surgery diagram for the Seifert-Weber space

In every reference I see, the Seifert-Weber space is presented as an identification space (specifically identify opposite faces of a dodecahedron by a 3/10ths twist).
What I can't seem to find is a ...

**2**

votes

**0**answers

70 views

### Criteria to decide whether a subset of the boundary complex of a polyhedron is a manifold?

Let $\mathcal{P} \subseteq \mathbb{R}^d$ be a convex polyhedron. Let $K$ be a subset of the boundary complex of $\mathcal{P}$. (Perhaps $K$ could be defined in terms of a system of linear ...

**3**

votes

**0**answers

109 views

### Is a generic link diagram semi-adequate?

Each crossing in a link diagram of a link $L$ has an A-resolution and a B-resolution.
Resolving all crossings gives a collection of circles in the plane, connected by dotted lines. A state of a ...

**2**

votes

**0**answers

124 views

### Discrete Isoperimetric Problem in the Hyperbolic Plane

Let $S_{g}$ be the closed orientable genus $g$ surface. I'm interested in studying a certain class $\Gamma$ of graphs on $S_{g}$ which fill (the complementary regions are simply connected), and which ...

**3**

votes

**1**answer

304 views

### On definition of surgery [closed]

I am a beginner in surgery theory. I have started learning with ALGEBRAIC AND GEOMETRIC SURGERY by Andrew Ranicki.
On page 4 of the book he defines surgery :
Denition 1.2 A surgery on an ...

**2**

votes

**3**answers

422 views

### A simple and good reference on surgery theory

Can anyone help me to find a simple and good reference (a book, lecture notes or a website) for learning the surgery theory and its applications? I seek a reference together with many examples and ...

**6**

votes

**1**answer

509 views

### How metric is Riemannian geometry

Let $(M, g)$ be a finite-dimensional Riemannian manifold. It is well-known, that the Riemannian metric induce a metric on the manifold by
$$d(x, y) = \text{inf} \int_a^b \| \dot\gamma(t) \| \, ...

**7**

votes

**1**answer

253 views

### Reference request: Spin structures on surfaces and the spin mapping class group

I am looking for references on the following: Spin structures on surfaces, and particularly the spin mapping class group.
What is known about generating the spin mapping class group? Has anybody ...

**4**

votes

**1**answer

410 views

### sufficient conditions on Non-Haken manifolds

Is there an algorithm to detect the Non-Haken Manifold?
Or, is there a sufficient condition for a manifold to be
a non-Haken manifold? (off course, I hope that condition is not the ones in its
...

**5**

votes

**0**answers

107 views

### Is the dimension given by Klee trick ever sharp?

The Klee Trick allows one to find an $\mathbb{R}^m$ where two embeddings of same compact metric space have homeomorphic complements. More precisely, given two embeddings of a compact metric space $K$ ...

**11**

votes

**1**answer

272 views

### Does a self map from the wedge sum of two spheres have either a fixed point or a point of period 2?

Let $X$ be the wedge sum of two $2$-dimensional spheres and $f$ a continuous function from $X$ into $X$. Does $f$ have either a fixed point or a periodic point of order 2?
Thanks