**5**

votes

**0**answers

89 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

94 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

135 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

157 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

233 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

139 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

134 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

195 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

408 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

267 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 ...

**9**

votes

**1**answer

181 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

383 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

235 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, ...

**4**

votes

**1**answer

110 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

22 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

126 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

300 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

317 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

194 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

190 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

133 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

162 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

100 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

32 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

292 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

257 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?

**-4**

votes

**1**answer

171 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

85 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

323 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

105 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 ...

**6**

votes

**1**answer

261 views

### Construction of invariants of 4-manifolds with the Kirby calculus

I'm an undergraduate student, interested in the low dimensional topology, in particular, the 4-manifold theory.
I have a question.
In the knot theory, the Reidemeister moves play fundamental roles.
...

**2**

votes

**0**answers

161 views

### Extra large spherical joins

If $X$ and $Y$ are piecewise spherical complexes, then their spherical join $X * Y$ is CAT(1) if and only of $X$ and $Y$ are CAT(1) (see the appendix of the first Charney-Davis paper below). One of ...

**0**

votes

**1**answer

167 views

### group actions on fibre bundles

Suppose that we have a group $G$ acting on the spaces $E$ and $B$. Suppose moreover that we have fibre bundles $\xi$ and $\eta$ in the following commutative diagram
If $\xi$ is a trivial bundle, ...

**3**

votes

**0**answers

101 views

### Uniform continuity of length function on geodesic currents

I'm starting to study geodesic currents and I have a question concerning uniform continuity.
Let's take $S$ a closed surface of genus $g$ and $GC(S)$ the space of geodesic currents on $S$ (as it is ...

**10**

votes

**1**answer

332 views

### Wild half-line in a Euclidean space

Is there an $m$-dimensional simplicial complex $S$ with the following properties:
The cone over $S$ is homeomorphic to $\mathbb{E}^{m+1}$. Here $\mathbb{E}^{m+1}$ denoes the $(m+1)$-dimensional ...

**3**

votes

**0**answers

145 views

### Are all finitely generated subgroups of SL2 LERF? [closed]

Let $H \leq \mathrm{SL}_2(\mathbb{R})$ be a finitely generated
subgroup. Must $H$ be LERF?
A group $H$ is said to be LERF (locally extended residually finite), or subgroup separable, if its ...

**1**

vote

**0**answers

89 views

### contact structure on double branched covers of $S^3$

We can assign a natural contact structure to double branched cover of a transverse knot $k$ in the $3$-sphere with its standard contact structure $(S^3,\xi_{st})$, as described ...

**5**

votes

**1**answer

128 views

### lower bound for Perron-Frobenius degree of a Perron number

A Perron number is an algebraic number which is greater than one in absolute value and is greater than all of its Galois conjugates in absolute value as well. Lind's theorem states that any Perron ...

**7**

votes

**1**answer

160 views

### Subgroups of the mapping class group of a surface generated by Dehn twists

Let $G$ be the mapping class group of a surface of genus $g > 1$. Is it known for which positive integer $k$ one can find a subgroup $H$ of $G$ generated by a finite number of Dehn twists and a ...

**8**

votes

**2**answers

218 views

### Quasi-isometric rigidity of Nil

Let $Nil$ be the unique simply connected non-abelian three-dimensional nilpotent Lie group, i.e. the group of upper triangular matrices with all the eigenvalues equal to 1 (this group is also known as ...

**13**

votes

**3**answers

658 views

### Classification of knots by geometrization theorem

I read this interview with Ian Agol, where he says:
"...I learned that Thurston’s geometrization theorem allowed a complete and practical classification of knots."
My question is:
How does ...

**6**

votes

**0**answers

82 views

### Survey of known results on equivariant transversality

Most basic differential topology theorems carry over to the equivariant case with mild modifications; see for instance Wasserman's paper. One thing that fails (more or less obviously) is equivariant ...

**3**

votes

**1**answer

139 views

### cohomology module of unit tangent vector bundles over spheres

Let $S^m$ be the $m$-sphere and $\tau (S^m)$ the sphere bundle consisting of unit tangent vectors in the tangent bundle $TS^m$. Then we have a fibration
$$
S^{m-1}\longrightarrow ...

**6**

votes

**2**answers

499 views

### quotient space of Eilenberg-MacLane space

Let $\pi$ be a group and $K(\pi,1)$ the Eilenberg-MacLane space. Let $G$ be a finite group acting on $K(\pi,1)$ such that the following is a covering map
$$
K(\pi,1)\longrightarrow K(\pi,1)/G.
$$
...

**4**

votes

**2**answers

251 views

### homotopy equivalence between configuration spaces

Let $M$ be a $m$-dimensional compact manifold without boundary and $W(M)$ the non-compact $CW$-complex obtained by glueing $[0,1)\times (0,1)^{m-1}$ to $M$, identifying the boundary ...