**0**

votes

**1**answer

107 views

### Maps of balls with fixed value along boundary

Suppose I wish to find the homotopy classes of maps of $B^3 \rightarrow M$ which along the boundary are fixed by a (particular) map $f: S^2 \rightarrow M$. Take $M$ to be a closed orientable ...

**1**

vote

**0**answers

140 views

### The fundamental group of the complement of codimension 2 submanifold

Suppose that $M^n$, $V^{n+2}$ are connected, compact smooth manifolds.
Let $f\colon M^n\to V^{n+2}$ is a smooth embedding.
Let $K_f$ be the kernel of the inclusion induced homomorphism ...

**2**

votes

**0**answers

199 views

### Fixed points of self maps

Given $m$ points on $S^n$, is there an explicit polynomial self $1-1$ map of minimum degree $f:S^n\rightarrow S^n$ that fixes only these $m$ points? Can we say something about symmetry group of $f$ if ...

**31**

votes

**1**answer

895 views

### Are there only countably many compact topological manifolds?

Up to homeomorphism, there are 2 one-dimensional topological manifolds and countably many 2- and 3-dimensional compact manifolds, respectively, since each manifold in these dimensions can be ...

**9**

votes

**1**answer

222 views

### regular tiling of a surface of genus 2 by heptagons

Let $S$ be a Riemann Surface of genus 2. Is there a picture in the literature for a regular tiling of $S$ by 12 heptagons (where three heptagons meet at each vertex). Also, apart from the obvious ...

**5**

votes

**1**answer

305 views

### Geodesics on manifolds with boundary

Let $(M,g)$ be a Riemannian manifold with non-empty boundary. Is there any notion of injectivity radius on $(M,g)$ in points away from the boundary? By this I mean points lying in $M- \partial M$. ...

**3**

votes

**2**answers

312 views

### Is this a $C^0$ foliation of $\mathbb{R}^2$?

Let $f(x)=\frac{1}{\sin(\pi x)}$ for $x\in (0, 1)$ and let
$\Gamma=\left\{(x,f(x)): x\in (0, 1)\subset \mathbb{R}^2\right\}$ be its graph.
For any set $X\subset \mathbb{R}^2$ and $\lambda>0$ and ...

**0**

votes

**0**answers

64 views

### discrete group action on Stiefel manifold

Let $S_3$ be the permutation group of order $3$. Let $V_2(\mathbb{R}^n)$ be the stiefel manifold of $2$-frames in $\mathbb{R}^n$. Let $S_3$ act on $V_2(\mathbb{R}^n)$ by
$$
(1,2)(u,v)=(-u,v-u),
$$
$$
...

**23**

votes

**2**answers

1k views

### fake $S^{2k}\times S^{2k}$

Let $X$ be a fixed closed manifold,$S(X)$ the structure set and $Aut(X)$ the group of self homotopy equivalence of $X$.
surgery theory tells us that $\mathcal{M}(X):=S(X)/Aut(X)$ is in bijection ...

**5**

votes

**1**answer

185 views

### Symmetric L-groups of integral group ring of finite cyclic groups

Where can i find the results about $L^{\ast}(\mathbb{Z}\pi)$ for $\pi$ a finite cyclic group?

**11**

votes

**1**answer

312 views

### What is obstructing two stably-isomorphic vector bundles from being isomorphic?

The specific situation is the following:
Let $n>0$ be a natural number, let $X$ be a finite CW complex of dimension $n$, and let $\xi_0,\xi_1$ be oriented real vector bundles of rank $n$ over $X$ ...

**1**

vote

**0**answers

47 views

### Local section of Lie Groupoids

Suppose we have the pair groupoid $G:\mathbb{R}^2\rightrightarrows \mathbb{R}$ which is a Lie groupoid with source $s$ and target $t$ maps given by the first and second projection, respectively. ...

**2**

votes

**2**answers

173 views

### Minimal genus of Seifert surface of torus knot

Let $(p,q)$ be a pair of coprime (positive) integers. Consider the torus knot $T_{p,q}$. What is the minimal genus of an (embedded) oriented Seifert surface for this knot?
It is not had to convince ...

**2**

votes

**1**answer

162 views

### Oriented knot complement conjecture for fibered knots

Suppose I have two inequivalent fibered knots in a homology sphere. When I say 'inequivalent', I mean that there is no orientation-preserving homeomorphism of the space that takes one to the other. ...

**2**

votes

**0**answers

119 views

### Piecewise smooth boundary version of Schoenflies Theorem

It is known that given any smooth ($C^\infty$) simple closed curve $\gamma$ in the 2-sphere, there is a smooth diffeomorphism of $\mathbb{S}^2$ taking $\gamma$ to the standard equator $x^2+y^2=1$.
...

**7**

votes

**1**answer

223 views

### Can two fibered knots have the same exterior?

Suppose I have two distinct fibered knots in a homology sphere. Is it possible for them to have (orientation-preservingly) homeomorphic exteriors?
See Oriented knot complement conjecture for ...

**1**

vote

**0**answers

152 views

### Pontryagin class of quaternionic line bundle

Let $\xi^{\mathbb{C}}$ be a complex line bundle over a CW complex $B$. Then
$$
VB_{\mathbb{C}^1}(B)\cong [B,BU(1)]=[B,\mathbb{C}P^\infty]=[B,K(\mathbb{Z},2)]\cong H^2(B;\mathbb{Z}).
$$
Hence if ...

**10**

votes

**3**answers

341 views

### Which polygons have *simple* periodic billiard paths?

I know (or, rather, believe) that it remains unknown whether every polygon
has a periodic billiard path.
But Masur proved in the 1980's that every rational polygon
(vertex angles rational multiples ...

**4**

votes

**0**answers

141 views

### When does a manifold 'deformation retract' into a small neighborhood of some k-dimensional subpolyhedron?

Under which conditions does a m-manifold $M^m$ admit a deformation retraction to a
small neighborhood of some k-dimensional subpolyhedron?
Or, under which conditions is the identity map $id_M$ of a ...

**0**

votes

**0**answers

141 views

### obstructions of Chern class and Pontryagin class

Let $\xi$ be a real $n$ dimensional vector bundle over a CW-complex $B$. Then the Stiefel-Whitney class (coefficient in $Z/2$)
$$
w_i(\xi)=0$$
if and only if $\xi|_{sk^i(B)}$ has $n-i+1$ linearly ...

**1**

vote

**0**answers

80 views

### Regularity of the taut foliation

In Eliashberg-Thurston's famous paper "Confoliations" Corollary 3.2.11, they proved that Irreducible three manifold with $b_{1}>0$ admits semi-fillable contact structure using Gabai's theorem in ...

**6**

votes

**2**answers

340 views

### Making spheres shellable

This is equivalent to my earlier question A question about something like "shelling" in a PL manifold, but maybe more comprehensible and to the point.
Given a triangulation of the PL sphere ...

**4**

votes

**2**answers

227 views

### Geometrisation of inclusion-like epimorphisms to free groups

Let $H_g$ be the standard $3$-dimensional handle-body, whose boundary is denoted $S_g$, the oriented closed surface of genus $g\geq 1$.
Call $F_g$ be the free group of rank $g$.
Denote by $i:S_g \to ...

**10**

votes

**1**answer

219 views

### Asymptotic dimension of $C'(1/6)$ small cancellation groups

Do there exist finitely presented $C'(1/6)$ small cancellation groups with arbitrarily high asymptotic dimension?
To offer a little more motivation, Roe proves that all hyperbolic groups have finite ...

**0**

votes

**1**answer

136 views

### Ambient isotopy of the diagonal submanifold in product space

Given a closed manifold $M^n$ and its $k$-fold product space $M^n\times\cdots\times M^n$,Can the diagonal submanifold $\Delta:=\{(m,\cdots,m)\in (M^n)^k\mid m\in M\}$ be isotopied to the submanifold
...

**1**

vote

**2**answers

332 views

### integral or rational cohomology of real grassmannians

I have obtained that the cohomology rings
$$
H^*(G_k(\mathbb{R}^\infty);\mathbb{Z}_2)=\mathbb{Z}_2[w_1,\cdots,w_k].
$$
Also
$$
H^*(G_k(\mathbb{R}^m);\mathbb{Z}_2)=\mathbb{Z}_2[w_1,\cdots,w_k]/(\bar ...

**9**

votes

**1**answer

346 views

### Reference request: Topology on the space of smooth compact submanifolds

In Allen Hatchers short exposition of the Madsen-Weiss Theorem he defines the topology on the space $\mathcal{C}^n$ of smooth oriented properly embedded $d$-dimensional submanifolds of $\mathbb{R}^n$ ...

**4**

votes

**0**answers

102 views

### Connected sum of chiral manifolds

Let $M,N$ be two closed, smooth, orientable manifolds of the same dimension and assume that these manifolds are chiral, i.e. they do not admit an orientation reversing automorphism. Then there are two ...

**8**

votes

**0**answers

134 views

### Closed geodesics avoiding points in hyperbolic surfaces

Let $\Sigma$ be a closed hyperbolic surface. Is it true that for any finite collection of points $x_1,\ldots,x_n\in\Sigma$ there exists a closed geodesic $\gamma$ containing none of them?
Remark: It ...

**3**

votes

**0**answers

80 views

### Geometric automorphism of free group respect to nonorientable suface

An outer automorphism $[ϕ]\in Out(F_n)$ is geometric if it is induced by a surface homeomorphism h:M→M, where M is a compact surface with nonempty boundary. I am wondering is it enough we only ...

**6**

votes

**1**answer

109 views

### What is the original reference for disorientations on tangle diagrams?

There are several invariants whose "natural" domain is a category of disoriented tangles, that is tangles which are piecewise-oriented, but which contain points called `disorientations' at which the ...

**8**

votes

**0**answers

183 views

### Alexander polynomial in branched covers

Suppose I am given a 3-manifold as a double branched cover over a link. Let a null-homologus knot in this space be given as a lift of an arc with endpoints on the link (it is automatically ...

**16**

votes

**1**answer

521 views

### Is every degree 1 self-map a homotopy equivalence?

In a rather obscure article, I found (without proof) the following statement:
If $M$ is a closed orientable manifold, every degree $1$ map $f: M \rightarrow M$ is a homotopy equivalence.
Is this ...

**3**

votes

**1**answer

148 views

### Braid wiring diagrams and matroids

recently I started reading some articles about the presentation of the fundamental group of lines arrangements in $\mathbb{C}^{2}$ via Wiring diagrams.
I also found some relation with matroid theory. ...

**1**

vote

**1**answer

330 views

### Isomorphism between a mapping class group and the fundamental group of a moduli space

For some fixed integer $d \geq 3$, let $M(0, d)$ be the mapping class group of self-homeomorphisms of the Riemann sphere which fix each of the $d$ points $0, 1, ... , d-2, \infty$. Let $X$ be the ...

**4**

votes

**1**answer

218 views

### Are “Unions” of small exotic $\mathbb{R}^4$'s small?

Suppose $M$ is a smooth 4-manifold, and $U,V \subset M$ are exotic $\mathbb{R}^4$'s, i.e. homeomorphic to standard $\mathbb{R} ^4$.
Further more suppose $U$ an $V$ intersect nicely sucht that $U \cup ...

**11**

votes

**1**answer

308 views

### Thom conjecture in CP3

Thom conjecture, that was originally asked in $\mathrm{CP}^2$, and is now proven for symplectic 4 manifolds, states that complex curves (symplectic surfaces) are genus minimizing in their homology ...

**6**

votes

**2**answers

287 views

### Is every $S^3$ block bundle over $S^4$ a fiber bundle?

I am interested in the difference between block bundle and fiber bundle.
Let $K$ be a simplicial complex and $p: E\to |K|$ be a continuous map.
A block diffeomorphism of $\Delta^p\times M$ is a ...

**1**

vote

**1**answer

104 views

### Convex subcomplexes of CAT(0) cubical complexes

Is the following statement true? If so, can anyone provide a reference?
Let $X$ be a CAT(0) cubical complex, and let $Y$ be a connected
subcomplex of $X$. Then the following are equivalent:
...

**11**

votes

**1**answer

356 views

### Is there a faithful transitive locally finite action of the modular group?

Is there a faithful transitive action of $G = \mathrm{PSL}_2(\mathbb{Z})$ on $\mathbb{Z}$ such that orbits under each $g \in G$ are finite?

**9**

votes

**2**answers

316 views

### How many knots are there with hyperbolic volume less than a given constant

Are there any known upper bounds on:
$$\#\left\{\text{hyperbolic knots }K\subseteq S^3\middle|\operatorname{Vol}(S^3\setminus K)<M\right\}$$
? I expect this grows at least exponentially in $M$, ...

**2**

votes

**2**answers

223 views

### F.g group with infinite ends not Q.I to a free group

Is there any easy example of a finitely generated group with a Cantor set of ends that is not quasi-isometric to a finitely generated free group?
Thanks in advance.

**15**

votes

**1**answer

553 views

### Smooth 4-manifolds with $E_8$ intersection form

Does there exist a closed orientable smooth 4-manifold $M$ whose intersection form is the $E_8$-form? Here by the intersection form I mean the $\mathbb{Z}$-valued bilinear form on ...

**5**

votes

**1**answer

381 views

### Are there spaces in which there are no fibered knots?

I am looking for orientable closed 3-manifolds in which there are no fibered knots. Although I know little about this, I think for links the answer to the question above is "no", and the result is ...

**2**

votes

**0**answers

159 views

### 3-manifold rigidity?

Defintion:
a $n$-manifold $M$ is said rigid if any homotopy equivalence $M\rightarrow N$ is homotopic to a homemorphism, where $N$ is an $n$-manifold.
The sphere $S^{3}$ and hyperbolic compact ...

**7**

votes

**1**answer

187 views

### Does every embedded 2-sphere in $\mathbb{R}^n$ bound an embedded ball?

Fix $n \geq 3$ and let $S \subset \mathbb{R}^n$ be a smoothly embedded $2$-sphere. Must there exist a smoothly embedded $3$-ball $B \subset \mathbb{R}^n$ such that $\partial B = S$? This is true for ...

**2**

votes

**0**answers

166 views

### Exotic actions of hyperbolic groups

Let $G$ be a hyperbolic group acting faithfully on $\mathbb{Z}$ such that:
The action is highly transitive - it is $k$-transitive for each $k \in \mathbb{N}$.
For every quasiconvex subgroup $H \leq ...

**4**

votes

**0**answers

111 views

### Sampling from a Manifold

Suppose we were to obtain a uniform sample, $S=\{x_1,...,x_m\}$, of points on a closed Riemannian $n$-manifold $M$. Let $\Gamma(S)$ be the set of all geodesics between the points in $S$ and we are ...

**10**

votes

**0**answers

198 views

### Is the quotient map of the action of homeomorphisms on embeddings well-behaved?

It is well known that if $M$ and $N$ are smooth manifolds, the diffeomorphisms $Diff(M)$ act continuously on the smooth embeddings $Emb^{C^\infty}(M,N)$ by precomposition, if both are given the ...

**3**

votes

**1**answer

309 views

### Knots in 3-manifolds

Consider a closed $3$-manifold $M$ and a knot $K$ in $M$.
Is it necessarily true that $\pi_2 (M \setminus K) = 0$?
If not, are there any conditions on $M$ and/or $K$ to ensure the above 2nd homotopy ...