**2**

votes

**0**answers

41 views

### cross-sections of a sphere bundle

Let $M$ be a $m$-manifold and $M_0$ a submanifold of $M$. Let $X$ be a pointed topological space. In the paper On the homology of configuration spaces, Bodigheimer-Cohen-Taylor, Topology 1989, ...

**25**

votes

**7**answers

3k views

### What are the open subsets of $\mathbb{R}^n$ that are diffeomorphic to $\mathbb{R}^n$

Hello,
I would like to know if there is a known necessary and sufficient
property on an open subset of $\mathbb{R}^n$ such that it is diffeomorphic to $\mathbb{R}^n$ :
For example :
1) Are all ...

**0**

votes

**1**answer

21 views

### Existenc conditions of single crossing [on hold]

There are two density function $f(x)$ and $g(x)$.They have the commnon support set $[\underline{x},\bar{x}]$.
Condition
(1)$f(\underline{x})<g(\underline{x})$ and $f(\bar{x})>g(\bar{x})$.
...

**6**

votes

**1**answer

183 views

### Is every nonabelian finite simple group a quotient of a triangle group $(a,b,c)$ with $a,b,c$ coprime?

Here by triangle group $(a,b,c)$ I mean the group with presentation
$$\langle x,y \;|\; x^a = y^b = (xy)^c = 1\rangle$$
In other words, for every finite simple nonabelian group $G$, do there exist ...

**23**

votes

**3**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 ...

**7**

votes

**1**answer

281 views

### Relation between moduli spaces and classifying spaces

I hope this question is suitable to be posted here on MO.
I wonder if there is a systematic relation between the notation of a classifying space, and the notion of a moduli space. I don't consider ...

**15**

votes

**1**answer

545 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 ...

**1**

vote

**0**answers

72 views

### How close (Homology-wise) can we approximate a topological manifold with a PL or smooth one?

Sorry if this question is to naive or badly phrased. I am curious about the following problem, given a manifold $M$, how "close" can we find a smooth or PL manifold, $N$, with a map $f:N\to M$. The ...

**17**

votes

**0**answers

143 views

### Metrics on the 3-sphere with knotted geodesics

According to answers to this question every metrics on $S^3$ admits a simple closed geodesic. Given a knot (or link) $K$, it's also quite simple to build a metric on $S^3$ such that $K$ is a geodesic ...

**4**

votes

**1**answer

116 views

### Counterexample to high dimensional Nielsen realization problem

Can someone give me an example of a closed smooth oriented manifold $M$ and an orientation preserving diffeomorphism $f:M\rightarrow M$ such that $f^k$ is isotopic to the identity for some $k\geq 1$, ...

**-6**

votes

**0**answers

72 views

### I need to know all geometry source to study geometry [closed]

I'm not specialist in geometry and I try to start learn all ph.d student need to know for their works. could you please introduce me which book should I start and please mention all book which is ...

**2**

votes

**1**answer

171 views

### Perturb a given smooth function to a Morse function relative to fixed level sets, which are already fine

This question was not answered on math.stackexchange.
Let $M$ be a manifold (not necessarily compact) , for the sake of clearness embedded in $\mathbb{R^n}$ and $f\colon M\rightarrow \mathbb{R}$ a ...

**5**

votes

**1**answer

328 views

### Homeomorphism of closed manifold

Suppose that we have two closed n-manifold $M$ and $N$ such that
the topological group of homeomorphisms $Homeo(M)$ is homotopy equivalent to $Homeo(N)$ (maybe as topological groups if needed), can ...

**6**

votes

**2**answers

194 views

### Equivariant smoothing of PL structures on $S^3$

Suppose $S^3$ is PL sphere on which a finite group $G$ acts by PL homeomorphisms. Is it always possible to find a compatible smooth structure such that $G$ acts by diffeomorphisms?
I am not quite ...

**2**

votes

**0**answers

60 views

### section spaces related to configuration spaces

In the paper Configuration spaces of positive and negative particles, Dusa McDuff, a section space $\Gamma(M)$ is constructed:
And in the paper ON THE HOMOLOGY OF CONFIGURATION SPACES. C.-F. ...

**3**

votes

**0**answers

40 views

### Determinant twist and $Pin _{\pm}$ structure on $4k$-dimensional bundles [Reference request]

Consider the automorphism $\varphi$ of $O(2n)$ given by $\varphi (A)=det(A)\cdot A$. The induced map in cohomology $H^*(BO(2n))$ sends $w_2$ to $w_2+(n+1)w_1^2$ (a proof is given at the end of the ...

**3**

votes

**2**answers

116 views

### Brieskorn homology spheres

We know that a Brieskorn homology 3-spheres $\Sigma(p,q,r)$ admit a free $S^1$-action, which makes it a Seifert fibered spaces with three singular fibers: $M(b;r_1,r_2,r_3)$. How should one get from ...

**2**

votes

**0**answers

92 views

### homotopy equivalence between configuration spaces on non-homeomorphic spaces

(1). Let $D^m$ be the closed $m$-disc in $\mathbb{R}^m$. For each $k$, does the $k$-th configuration space on $D^m$ homotopy equivalent to the $k$-th configuration space on $\mathbb{R}^m$
$$
...

**9**

votes

**0**answers

127 views

### Why should we regard $PL(M)$ as a simplicial group?

Let $M$ be a manifold. If $M$ is smooth, it is clear what $\text{Diff}(M)$ should be, namely it should be the set of diffeomorphisms of $M$ equipped with the topology in which a sequence of ...

**104**

votes

**2**answers

10k views

### What are the shapes of rational functions?

I would like to understand and compute the shapes of rational functions, that is, holomorphic maps of the Riemann sphere to itself, or equivalently, ratios of two polynomials, up to Moebius ...

**3**

votes

**1**answer

205 views

### Rigidity of secondary characteristic classes

For a representation $\rho:\pi_1M\rightarrow GL(n,C)$ and the associated flat $GL(n,C)$-bundle $E_\rho\rightarrow M$ one has the Cheeger-Chern-Simons classes
$$\hat{c}_k(E_\rho)\in H^{2k-1}(M,R/Z)$$
...

**41**

votes

**1**answer

4k views

### Open map D⁴ → S²

Is it possible to construct an embedding $D^4\hookrightarrow S^2\times
\mathbb R^2$
such that the projection $D^4\to S^2$ is an open map?
Here $D^n$ denotes closed $n$-ball.
An open map D⁴ → S².
It ...

**11**

votes

**1**answer

282 views

### Homology spheres and fundamental group

I have a curiosity about homology spheres: I was wondering if they were uniquely characterized by their fundamental group. I.e. given two $n-$dimensional (integral) homology spheres with isomorphic ...

**9**

votes

**1**answer

134 views

### bi-Lipschitz gluing

Let $H$ be the Heisenberg group with
left invariant sub-Riemannian metric and $\varepsilon>0$ is small.
Let us denote by $|x-y|_H$ the distance from $x$ to $y$ in $H$.
I have a bi-Lipschitz ...

**1**

vote

**1**answer

111 views

### Reference request: $\mathcal{C}^\infty_c(M)$ is a topological vector space with the Whitney topology

Let $M$ be a smooth manifold and let $\mathcal{C}^\infty(M)$ be the space of all smooth real valued functions with the (strong) Whitney topology. This space is a topological vector space iff $M$ is ...

**22**

votes

**6**answers

2k views

### Failure of smoothing theory for topological 4-manifolds

Smoothing theory fails for topological 4-manifolds, in that a smooth structure on a topological 4-manifold $M$ is not equivalent to a vector bundle structure on the tangent microbundle of $M$. Is ...

**-1**

votes

**0**answers

21 views

### Freedom of speech in scientific discussions - An invitation to more tolerance in Scientific debates [migrated]

I hope this post enjoys some tolerance, and don't get closed or put on hold immediately.
I believe that freedom of speech in scientific discussions is one of the key values which enriches the debates ...

**5**

votes

**1**answer

176 views

### Hyperbolic knot complement groups and relative dimension

Let $G$ be a the fundamental group of a hyperbolic knot complement.
Then $G$ is hyperbolic relative to a subgroup $P\cong \mathbb Z \oplus \mathbb Z$.
The knot complement has a $2$-dimensional spine ...

**5**

votes

**0**answers

65 views

### Complements of unknotted tori (higher dimensions)

It is weil-known that an unknotted 2-torus in $S^3$ provides the standard Heegaard splitting, in particular its complement consists of two solid tori.
It is also known that an unknotted 3-torus in ...

**1**

vote

**6**answers

754 views

### Good books on Geometric Theory of Dynamical Systems

I am looking for a good book on Geometric Theory of Dynamical Systems . I found Geometric Theory of Dynamical Systems by Jr. Palis myself,but it's very old, anyway i would like to find a pure ...

**5**

votes

**0**answers

170 views

### How to get a polygon from a translation surface $(X,\omega)$

Let $S_g$ be a compact topological surface of genus $g$. I know there is the correspondence
$\{$Abelian differentials on compact Riemann surfaces of genus g$\}\leftrightarrow\{$ Translation surfaces ...

**7**

votes

**2**answers

194 views

### Automorphism of genus 2 surface with 5 fixed points

Is there a self-homeomorphism of a genus 2 (closed, orientable) surface, which has finite order and exactly 5 fixed points?
Of course, the same question can be asked replacing 2 by $g$ and $5$ by any ...

**7**

votes

**2**answers

168 views

### What is the easiest way to compute Ozsváth-Szabó tau invariant of a knot?

Suppose that we have a knot $K$ with 40 crossings which is not a cable knot or an alternating knot.
Then, what is the easiest way to compute Ozsváth-Szabó's invariant $\tau(K)$?
Are there any ...

**19**

votes

**3**answers

2k views

### Riemann mapping theorem for homeomorphisms

How do you prove to any two simply-connected domains in the plane are homeomorphic without using the Riemann mapping theorem? An elementary proof would be nice.

**6**

votes

**1**answer

192 views

### Can a 3-ball divide a standard 4-ball into two exotic 4-balls?

Let $B^n$ denote the unit ball in $\mathbb{R}^n$ (wrt the standard euclidean metric) and $\bar{B}^n$ denote the unit closed ball. Suppose that $\Sigma$ is a a smooth embedded hypersurface with ...

**1**

vote

**0**answers

196 views

### On matrix rank inequality

Let $A$ be a $\{0,1\}$ square matrix.
Let $J$ be all $1$ matrix.
Let $\bar{A}=J-A$.
Is it possible for $rk_+(A)\geq c\cdot rk_+(\bar{A})^d-1$ and $rk_+(\bar{A})\geq c\cdot rk_+({A})^d-1$ for some ...

**3**

votes

**1**answer

88 views

### Involution on the set of isomorphism classes of 2n-dimensional vector bundles induced by an involution of $O(2n)$

Denote by $\varphi$ the automorphism of $O(2n)$ given by $\varphi (A)=det(A)\cdot A$.This induces a self-map $B\varphi$ of $BO(n)$, so it induces a self-map (actually an involution)
$B\varphi ^*$ on ...

**4**

votes

**1**answer

176 views

### Can a homology $n-1$-sphere divide $\mathbb{S}^{n}$ into non-contractible components?

This is a follow-up to my earlier question.
Let $\Sigma\subset \mathbb{S}^{n}$ be a hypersurface -- here $\mathbb{S}^{n}$ is a smooth sphere (possibly exotic ... if this makes a difference). If ...

**4**

votes

**1**answer

162 views

### Topology of hypersurface of sphere fixed by homeomorphic involution

I'm not an topologist, so I apologize in advance if this is a silly question.
I have the following situation, let $\mathbb{S}^n$ (for $n\geq 3$) be the standard (smooth if it matters) $n$-sphere and ...

**8**

votes

**0**answers

170 views

### Simplices and cubes

Question: What is the first appearance in the literature of one of the
following statements:
The result of intersecting a simplex with a cell of the dual
subdivision is a cube
There ...

**5**

votes

**0**answers

100 views

### Framed singular knots

I've recently run across what one might (and I suspect people probably do) call framed singular knots, or maybe singular ribbon knots. Regardless of the name, what I mean is the following: Let $D$ be ...

**4**

votes

**2**answers

179 views

### Exponential rule for Whitney-$\mathcal{C}^{\infty}$-topology

Let $M,N,X$ be smooth manifolds. Equip the space of smooth functions between two manifolds with the (strong) Whitney- $\mathcal{C}^\infty$-topology.
The evaluation map $$ev\colon ...

**7**

votes

**1**answer

290 views

### Z/p action on finite contractible complex

Let $p$ be a prime and $X$ a finite contractible CW-complex. Assume $\mathbb Z/p$ acts on $X$. Then it is easy to see that there has to be a fixed point. (E.g. use Lefschetz's fixed point theorem or ...

**3**

votes

**1**answer

215 views

### Fixed point property for intersection of spaces which are homeomorphic to a disk

The following question is question 9.8 from Miller's paper ``Some interesting problems
'':
Question Suppose $D_n$ a subset of the plane is homeomorphic to a disk and for every
$n\in \omega, ...

**6**

votes

**1**answer

91 views

### Reference for a PL flat torus embedding in $\mathbb{R}^3$

A piecewise linear flat torus embedded in $\mathbb{R}^3$ is shown at http://www.mathcurve.com/polyedres/toreplat/toreplat.shtml.
It is flat in the sense that the angle defect at the vertices is zero.
...

**37**

votes

**11**answers

4k views

### Compelling evidence that two basepoints are better than one

This question is inspired by an answer of Tim Porter.
Ronnie Brown pioneered a framework for homotopy theory in which one may consider multiple basepoints. These ideas are accessibly presented in his ...

**3**

votes

**1**answer

230 views

### Simple proof for property R conjecture

Gabai's property R theorem is:
If the 0-surgery manifold of a knot $K$ is homeomorphic to $S^1\times S^2$, then $K$ is the unknot.
Recently, 3-manifold topology has been developed rapidly by Agol, ...

**7**

votes

**2**answers

512 views

### Embedded (framed) cobordisms

[The title initially was "Actions of gauge groups on framed cobordisms. This has been changed.]
This question is a follow-up to my answer to When is a submanifold of $\mathbf R^n$ given by global ...

**0**

votes

**0**answers

80 views

### Codimension one embeddings

For smooth knots in $S^3$ "isotopy" and "ambient isotopy" are equivalent (although this is not true in the topological category). I guess that therefore also for tori in $S^3$ "isotopy" and "ambient ...

**13**

votes

**1**answer

165 views

### Three-manifolds having a Reebless foliation but not a taut one

A straightforward argument reveals that a taut foliation is Reebless, and of course there are many examples of Reebless foliations that are not taut. I guess that there are many examples of ...