**0**

votes

**2**answers

109 views

### obtaining circle bundle over torus by trefoil surgery

Does any integer surgery on a right or left trefoil knot give the $S^1$-bundle over $T^2$ with Euler number $1$?

**0**

votes

**0**answers

97 views

### Mapping class groups acting on simple closed curves

Let $S_{g,d}$ be a genus $g$ compact Riemann surface with $d$ punctures. Let $\mathcal{M}_{g,d}$ be the moduli space of all such surfaces, i.e. genus $g$ compact Riemann surfaces with $d$ marked ...

**2**

votes

**0**answers

61 views

### Singularities of Families of Differential Equations

Suppose you are given a family of differential equations with rational coefficients: $\partial_x^ny+a_1(x,x_1,\dots,x_m)\partial_x^{n-1}y+\dots+a_n(x,x_1,\dots, x_m)y=0$. At each point $X=(x_1,\dots, ...

**8**

votes

**1**answer

185 views

### Homotopy type of diffeomorphism which are the identity on and near the boundary

Let $M$ be a compact manifold with boundary. Denote by $Diff(M), Diff_\partial(M)$ and $Diff_{U\partial}(M)$ the groups of diffeomorphisms of $M$ and the subgroups of the ones that are the identity on ...

**4**

votes

**0**answers

139 views

### Questions on Thurston's metric on Teichmüller space

I'm reading the famous "Minimal stretch maps between hyperbolic surfaces" by William Thurston and I'm trying to understand the key theorem 8.1. I have many unclear points so I hope someone can help me ...

**6**

votes

**3**answers

255 views

### For an arithmetic hyperbolic 3-manifold group, when is its trace field not its invariant trace field?

Edit: In my original post I failed to require the group to be a manifold group. The answer below from @BenLinowitz works in that case. I am really interested though in when the group is torsion-free, ...

**4**

votes

**1**answer

253 views

### Who first considered constructibility of simplicial complexes?

A simplicial complex of dimension $d$ is called constructible if it is a simplex, or if it is the union of two constructible dimension-$d$ simplicial complexes along a dimension-$(d-1)$ intersection. ...

**8**

votes

**1**answer

123 views

### Example where Calabi invariant is nontrivial?

Let $D^2$ denote the closed unit disk in $\mathbb{R}^2$. Let $\omega := dx \wedge dy$ denote the standard area form on $\mathbb{R}^2$ (and on $D^2$ by restriction). Let $\phi$ be a diffeomorphism of ...

**6**

votes

**0**answers

131 views

### kernel of the mod $2$ Bockstein on the first cohomology group

Let $M$ be a path-connected finite $CW$-complex. Suppose the first integral homology group is $H_1(M;\mathbb{Z})= \mathbb{Z}_2^{\oplus r}\oplus A$ where $r\geq 1$ and $A$ is a finite abelian group of ...

**3**

votes

**1**answer

56 views

### vector bundles induced by an action of a finite subgroup of $O(n)$

Let $M$ be a path-connected manifold. Let $G$ be a finite subgroup in $O(n)$ and suppose $G$ acts freely on $M$. Then we have an associated vector bundle
$$
\xi(M,G): \mathbb{R}^n\longrightarrow ...

**2**

votes

**0**answers

110 views

### Connecting homomorphism in generalized cohomology theory

I have some compact manifold with boundary $(M,\partial M)$, and there is a long exact sequence
$$\cdots\to KO^{-1}(\partial M)\xrightarrow{\partial} KO^{0}(M,\partial M)\to KO^0(M)\to KO^0(\partial ...

**1**

vote

**1**answer

51 views

### Existence of a continuous map of a disk with a given boundary image on a surface to its complement in $\mathbb{R}^3$

I am considering the following problem:
given an embedded closed surface in $\mathbb{R}^3$ (unknotted) and a non-trivial simple closed curve on it, does there exist a continuous map of a disk to the ...

**5**

votes

**0**answers

92 views

### cohomology ring of configuration spaces on $S^2$ and the projective plane

For a manifold $M$ and a positive integer $n$, the unordered configuration space $B(M,n)$ is the space consisting of all unordered collections of $n$ distinct points on $M$. Precisely,
$$
...

**6**

votes

**1**answer

134 views

### non-orientability of vector bundles induced from a symmetric group action

Let $\Sigma_k$ be the symmetric group on $k$-letters. Let $M$ be a manifold with a free $\Sigma_k$-action. Then we can form a $k$-dimensional vector bundle
$$
\xi:\mathbb{R}^k\longrightarrow ...

**16**

votes

**2**answers

785 views

### What is an example of an orbifold which is not a topological manifold?

In Thurston's book The Geometry and Topology of Three-Manifolds it is proven that the underlying space of a two-dimensional orbifold is always a topological surface.
Are there any easy examples of ...

**5**

votes

**0**answers

100 views

### Mal'cev completions of finitely generated torsion-free nilpotent groups

There is some question from geometric group theory:
One wonders if the following conditions are equivalent for finitely generated torsion-free nilpotent groups $\Gamma$ and $\Lambda$:
$\Gamma$ and ...

**0**

votes

**0**answers

98 views

### Triangulation of S^2xS^2

Could someone tell me or give a reference for the minimal triangulation of $S^2\times S^2$ and $S^2\times S^1\times S^1$ ?
Thanks,

**3**

votes

**1**answer

139 views

### Euler Class constant on Fibered Face of Unit Thurston Norm Ball?

I am reading about the Thurston norm out of Candel and Conlon's "Foliations 2" and Calegari's "Foliations and the Geometry of 3-Manifolds".
I'm trying to work through the proof of the "Fibered ...

**5**

votes

**1**answer

162 views

### Is every triangulation of a Euclidean ball by convex tetrahedra shellable?

Suppose you are given a 3-ball $B$ in $\mathbb{R}^3$ that is bounded by a PL sphere, a triangulation $T$ of $B$ by Euclidean tetrahedra. Is that triangulation necessarily shellable?
I know that if ...

**10**

votes

**1**answer

238 views

### Closed $3$-manifold, $2$-dimensional subbundle of this manifold, is this form exact or not?

Let $M$ be a closed $3$-manifold, and let $\xi$ be a $2$-dimensional subbundle of $TM$. I know the following.
There is a nowhere zero $1$-form $\alpha$ on $M$ with $\alpha(X) = 0$ for any vector ...

**4**

votes

**1**answer

145 views

### mapping class group of a two-holed torus

It is well-known that in the picture below we have $t_d=(t_at_b)^6$ as elements in the mapping class group of a two-holed torus, ($t_\gamma$ represents positive Dehn twist about the curve $\gamma$). ...

**4**

votes

**1**answer

136 views

### Smooth closed simply-connected $4$-manifold with $w_1 = w_2 = 0$ without a point admits symplectic structure?

See my previous question here.
Let $M$ be a smooth closed simply-connected $4$-manifold with $w_1 = w_2 = 0$. Can $TM$ be trivialized in the complement of a point?
This was answered in the ...

**7**

votes

**2**answers

137 views

### Tangent bundle of smooth closed simply-connected $4$-manifold $w_1 = w_2 = 0$ can be trivialized in complement of point?

Let $M$ be a smooth closed simply-connected $4$-manifold with $w_1 = w_2 = 0$. Can $TM$ be trivialized in the complement of a point?

**8**

votes

**1**answer

196 views

### Is the action of $G$ on $H_1(T^n, \mathbb{Z}) = \mathbb{Z}^n$ faithful?

Let $G$ be a finite group of diffeomorphisms of the torus $T^n$ fixing some point $p$, i.e. $p$ is fixed by every element of $G$. I have two questions.
Is the action of $G$ on $H_1(T^n, \mathbb{Z}) ...

**3**

votes

**0**answers

70 views

### Is there a Riemannian metric on the $2$-ball $B^2$ which has negative sectional curvature everywhere, and such that the boundary is concave?

Is there a Riemannian metric on the $2$-ball $B^2$ which has negative sectional curvature everywhere, and such that the boundary is concave; i.e. the geodesic curvature along the boundary points ...

**1**

vote

**0**answers

79 views

### coefficient of homology of configuration spaces over real projective spaces

In the slides Characteristic Classes of Surface Bundles
and Configuration Spaces, Miguel A. Xicot'encatl, page 38, what is the coefficient of the following homology?
Could the coefficient be an ...

**4**

votes

**2**answers

410 views

### Generalized Jordan theorem and winding number

By the generalized Jordan theorem any continuous injective map
$S^{n-1} \hookrightarrow R^n$ splits $R^n$ into two regions, one being bounded (interior) and the other one unbounded (exterior). It ...

**4**

votes

**1**answer

272 views

### What are the 4 convex simplicial 4-polytopes that have 6 vertices?

In Convex polytopes and related complexes by Klee and Kleinschmidt they state the number of $d$-polytopes with $d+2$ vertices is $\lfloor \frac{d^2}{4}\rfloor$.
I was wondering what the four ...

**8**

votes

**1**answer

183 views

### Dimension in Whitney's theorem

There are some interesting examples of (classes of) manifolds for which I suspect that the Whitney theorem can be strenghtened. For example it is known that every (smooth) $n$-dimensional manifold can ...

**11**

votes

**3**answers

388 views

### $A_{\infty}$-structure on closed manifold

Is there an exmaple of a closed smooth connected manifold $M$ having a structure of $A_{\infty}$-space (with unit) but $M$ is not homeomorphic to a compact connectd Lie group as space ?
Edit: First, ...

**13**

votes

**1**answer

250 views

### Do vanishing characteristic classes of the tangent bundle imply a manifold is stably frameable?

Suppose we know that all stable characteristic classes of the tangent bundle of a manifold $M$ vanish, i.e. the map $f:M\rightarrow BO$ stably classifying the tangent bundle is trivial on cohomology, ...

**5**

votes

**1**answer

117 views

### boundary of semihyperbolic groups

There are various definitions of boundary of a hyperbolic group. Which of those generalize to semi-hyperbolic groups (in the sense of Alonso and Bridson)?
The example I have in mind is a semisimple ...

**1**

vote

**0**answers

24 views

### Homology of the subcomplexes of the “diamond shaped” sphere under 1-norm in $R^n$ as a simplicial complex

The 1-norm on $\mathbb{R}^n$ is defined by $\|v\| = |v_1| + |v_2| + \cdots + |v_n|$ for a vector $v = (v_1, \ldots, v_n) \in \mathbb R^n$.
The unit sphere $S^{n-1}_1$ under the 1-norm is a simplicial ...

**6**

votes

**1**answer

129 views

### Maximal TB number and slice genus relation of a knot in any 3-manifold

Lisca-Stipsicz say that if a knot $K\subset{S^{3}}$ satisfies $g_{s}(K)>{0}$ and $TB(K)=2g_{s}(K)-1$ then $S^{3}_r(K)$ carries positive, tight contact structures
for every ...

**16**

votes

**1**answer

306 views

### Is the restriction map for embeddings of manifolds with boundary a fibration?

Let $M$ and $W$ be smooth manifolds (possibly with boundary) and $V\subseteq W$ a submanifold. We have a map between embedding spaces
$$Emb(W,M)\rightarrow Emb(V,M)$$ given by restriction.
Richard ...

**10**

votes

**2**answers

321 views

### Stable homotopy groups of $RP^{\infty}$

Are the stable homotopy groups $\pi^s_i(\mathbb R P^{\infty})$ known for small $i$? In particular, I would be interested in the values for $i = 5,6$. A quick Internet search did not lead to anything.

**4**

votes

**0**answers

203 views

### A generalized ellipse

We know that an ellipse is the locus of all point $z$ in the plane with $$|z-a|+|z-b|=\lambda$$
where $a,b$ are two given points in the plane and $\lambda$ is a constant.
Now we consider the ...

**4**

votes

**1**answer

191 views

### Local product structure of determinantal variety

The variety $X_n$ of singular $n\times n$ real matrices is stratified by smooth strata $X_{n,k}$ where $k$ is the rank. Choose a rank $k$ matrix $A\in X_{n,k}$. Is there a local diffeomorphism sending ...

**7**

votes

**1**answer

139 views

### Admissibility in Heegaard Floer, especially with torsion Spin^c structures

I'm confused about the relationship between strong admissibility and weak admissibility for pointed diagrams in Heegaard Floer theory. For reference, here are Ozsváth-Szàbo's original definitions:
...

**8**

votes

**1**answer

172 views

### Simply connected noncompact surfaces

Is there a theorem saying that any noncompact, simply connected topological surface is homeomorphic to the plane ? There seems to be many well-known results about the classification of compact ...

**3**

votes

**0**answers

105 views

### Intersection patterns of loops on surfaces

Let $a,b$ be to simple closed loops on a surface $S$ with homologically trivial intersection (more generally I'm also interested in the case when $b$ is 1-codimensional). Denote their intersection on ...

**0**

votes

**0**answers

34 views

### Approximation of a volume-preserving Hölder homeomorphism by diffeomorphisms?

Is it known whether a volume-preserving Hölder homeomorphism of an arbitrary manifold can be approximated by a volume-preserving diffeomorphism?
The answer is clearly no if the volume-preserving ...

**10**

votes

**3**answers

296 views

### Random links and $3$-manifolds

In Jeffrey Weeks book "The Shape of Space" he explaines at the end of Chapter 18 (on page 255) the following about the geometrization conjecture:
A non-trivial connected sum $M_1\# M_2$ admits a ...

**0**

votes

**0**answers

45 views

**12**

votes

**1**answer

261 views

### Hyperbolic 3-manifold groups acting on the plane

Can the fundamental group of a closed hyperbolic 3-manifold act freely on the plane by homeomorphisms? Freely and cocompactly? Freely, cocompactly, and preserving orientation?

**-3**

votes

**1**answer

172 views

### Homeomorphism between (-1,1)×[-1,1) and [-1,1]×[-1,1) [closed]

Can one construct homeomorphism between (-1,1)×[-1,1) and [-1,1]×[-1,1)?
If so, please show me how to construct it.

**5**

votes

**1**answer

89 views

### stabilization of Legendrian knots

There are two ways to stabilize a Legendrian knot $k$ in standard contact sphere $(S^3,\xi_{st})$ i.e. adding right cusps or left cusps, let's call these two stabilized Legendrian knots $k_R$ and ...

**14**

votes

**0**answers

325 views

### pizza lemma (topology)

given six real-analytic arcs in the unit disk $D$, each of which
connects the origin to a boundary point, and no two arcs meet anywhere except
at the origin, and the arcs meet at equal (60 degree) ...

**1**

vote

**0**answers

138 views

### Braids with an infinite number of strings

Has anyone developed a theory for braids with an infinite number of strings?

**2**

votes

**1**answer

108 views

### Existence of non-negative extensions of smooth functions on axes

I am struggling to solve an extension problem of smooth functions, and I would like someone to help me.
The setting is as follows:
Let $X_1$, $X_2$, and $T$ be either the real lines $\mathbb R$ or ...