# Tagged Questions

**6**

votes

**1**answer

311 views

### Status of Zeeman's collapsability Conjecture

Zeeman's conjecture in topological combinatorics states that if $K$ is a contractible polyhedron of dimension 2, then $K\times I$ has a collapsible subdivision.
What is the status of this ...

**3**

votes

**0**answers

81 views

### When closed subsets have finitely many connected componenets

Let $X$ be topological space such that every its closed subset has finitely many connected componenets. Is there any charactrization for such topological space?

**2**

votes

**1**answer

104 views

### Why simple closed curves are dense in $\mathcal{PML}_0(S)$?

I have another question about laminations on surfaces. As usual let $\mathcal{S}$ be the set of homotopy classes of simple closed curves in $S$ and $\mathcal{PML}_0(S)$ be the set of projective ...

**3**

votes

**1**answer

122 views

### Why is $\mathcal{PML}_0(S)$ compact?

I'm starting to study geodesic laminations on hyperbolic surfaces and in particular I'm focusing my attention on $\mathcal{PML}_0(S)$, the space of projective classes of measured geodesic laminations ...

**1**

vote

**1**answer

119 views

### triviality of a $2$-sheeted covering map and the triviality of the associated vector bundle

Let $ X$ be a space with a (free and properly discontinuous) $\mathbb{Z}/2$-action and $$p: X\to X/(\mathbb{Z}/2) $$ be a $2$-sheeted covering map. Then we have an associated vector bundle
$$
\xi: \...

**3**

votes

**1**answer

200 views

### geometric conditions on maps between manifolds inducing monomorphisms on cohomology

Let $M,N$ be manifolds whose dimensions may be different. Let $f: M\longrightarrow N$ be a smooth map. What geometric conditions on $f$ can we impose such that the induced homomorphism
$$
f^*: H^*(N;\...

**6**

votes

**0**answers

115 views

### Connectedness of cones in the boundary of a 1-ended hyperbolic group

Let $G$ be a one-ended hyperbolic group. We can think of the boundary of $G$ as consisting of geodesic rays originating at the identity in some Cayley graph, modulo the relationship of being ...

**12**

votes

**1**answer

428 views

### Realizing symmetric groups by diffeomorphisms

Let $M$ be a (closed, smooth) manifold of dimension $d$. For $n$ a positive integer, fix $n$ points $x_1, \dots, x_n \in M$. The group of diffeomorphisms of $M$ that permutes the points $x_i$ surjects ...

**4**

votes

**0**answers

88 views

### Weakest topology on curves of $R^2$ that makes length a continuous function

We know that the length of curves is not semi-continuous for certain topologies, and is for some others. A natural question I asked myself recently is "what do we need to make length continuous", i.e.:...

**17**

votes

**1**answer

389 views

### Vector field on a K3 surface with 24 zeroes

In http://mathoverflow.net/a/44885/4177, Tilman points out that one can use a $K3$ surface minus the zeroes of a generic vector field to build a nullcobordism for $24[SU(2)]$. Given that a) this is a ...

**1**

vote

**0**answers

60 views

### Characterization of certain families of functions

For $R$ equal $\mathbb{R}$ or $\mathbb{Z}$, let $D^+_R:=\{(x,y)\in R^2\colon x<y\}$. For each natural $n$, let $F_{n,R}$ denote the set of all Borel-measurable functions $f\colon D^+_R\to\mathbb{R}$...

**1**

vote

**0**answers

71 views

### Existence of polytope

Does there exist a polytope in dimensional d consisting of $k>d+1$ faces satisfying that every d faces intersect? I tried 3 dimensional cases, and it seems negative. But is it all negative for any ...

**2**

votes

**1**answer

109 views

### positions of regular cubes in Euclidean space with all its vertices without distinction

Let $P$ be the standard regular cube, centered at the origin of $\mathbb{R}^3$ with all its vertices on the unit sphere.
If the vertices are labelled by $1,2,3,4,5,6,7,8$, then the collection of ...

**8**

votes

**1**answer

228 views

### references for Stiefel-Whitney class of Stiefel manifolds and Grassmannians

Let $M$ be a manifold. The total Stiefel-Whitney class of $M$ is defined to be the Stiefel-Whitney class of the tangent bundle $TM$
$$
w(M)=1+w_1(TM)+w_2(TM)+\cdots
$$
I want to find references for
$$
...

**6**

votes

**0**answers

177 views

### A tangential fixed point property for manifolds embedded in Euclidean spaces

Assume that $M$ is a compact orientable manifold which is embedded in some Euclidean space $\mathbb{R}^{N}$
We say that $M\subset \mathbb{R}^{N}$ has the tangential fixed point property if for ...

**8**

votes

**1**answer

171 views

### Conditions on the hierarchy for Thurston's hyperbolization theorem

From my understanding the proof of Thurston's hyperbolization theorem for Haken $3$--manifolds consists of cutting the manifold along a hierarchy (collection of incompressible, $\partial$-...

**3**

votes

**0**answers

129 views

### Is $C^\infty(M,\mathbb{R})$ an ind-(smooth manifold)?

Let $\mathrm{Man}$ be the category of smooth manifolds (2nd countable, Hausdorff, no boundary, not necessarily compact) and smooth maps, and let $M$ be an object thereof. Is the presheaf $S \mapsto C^...

**4**

votes

**1**answer

115 views

### Is the signature of inverse images of diffeomorphic submanifolds (along a homotopy equivalence) the same?

Suppose it is given an orientation preserving homotopy equivalence $h:N→M$ between closed oriented connected manifolds. Let $X,Y\subset M$ be diffeomorphic submanifolds, and assume $h$ to be ...

**6**

votes

**1**answer

257 views

### vector bundles associated to a covering space

Let $M$ be a $m$-dimensional manifold whose cohomology ring and cell structure are well-understood, such that there is a free action of the symmetric group $S_n$ on $M$. Then we have a $n!$-sheeted ...

**6**

votes

**1**answer

189 views

### Fundamental class in $KO[1/2]$

Let $M^m$ be an oriented Riemannian manifold.
The signature operator associates to $M$ a class $\Delta_M\in KO_m(M)[1/2]$.
I have two questions about this class $\Delta_M$:
Rationally, $\Delta_M$ is ...

**23**

votes

**6**answers

2k views

### How to get convinced that there are a lot of 3-manifolds?

My question is rather philosophical : without using advanced tools as Perlman-Thurston's geometrisation, how can we get convinced that the class of closed oriented $3$-manifolds is large and that ...

**4**

votes

**0**answers

95 views

### characteristic classes of a covering space with symmetric group action

Let $S_n$ be the $n$-th symmetric group. Suppose we have a $n!$-sheeted covering space
$$
S_n\to M\to M/S_n
$$
where $M$ is a manifold. Let $\mathbb{K}$ be the real numbers $\mathbb{R}$, complex ...

**7**

votes

**2**answers

212 views

### Heegard genus of hyperbolic Haken 3-manifolds

Is there an example of a closed Haken hyperbolic 3-manifold of Heegaard genus 2?

**3**

votes

**1**answer

487 views

### What is the Euler characteristic of a mapping space?

Suppose that $A$ and $B$ are topological spaces homotopy equivalent to finite cell complexes, and let $B^A = \mathrm{maps}(A,B)$ denote the space of maps from $A$ to $B$. Is it there a formula for ...

**2**

votes

**1**answer

161 views

### A question about simple closed curves in 3-dimensional Euclidean space

Let E(3) be 3-dimensional Euclidean space. I have submitted the following question to Mathstackexchange and other mathematical websites, but have never received any responses-not even rejections on ...

**10**

votes

**1**answer

490 views

### positions of a methane molecule with carbon atom at the origin

Let $\text{CH}_4$ be the molecule of Methane:
The four hydrogen atoms form vertices of a regular tetrahedron with the carbon atom in the center of the regular tetrahedron.
Here we regard all atoms ...

**10**

votes

**0**answers

212 views

### Mapping class groups in high dimension

Let $M$ be a $1$-connected, closed, smooth manifold with $dim(M)>4$ and let us set $MCG(M)=\pi_0(Diff(M))$. Dennis Sullivan proved that $MCG(M)$ is commensurable to an arithmetic group.
I was ...

**11**

votes

**1**answer

255 views

### Reference request: sheaves on the site of d-manifolds

I believe I know how to prove the following results. I also know to whom to cite fancy-shmancy results that have these as a very special case. My question is: what are the correct citations for ...

**9**

votes

**1**answer

201 views

### Inducing up the group homomorphism between mapping class groups

There are many ways to embed the braid group into the mapping class group of a surface. To describe one of them, let ${C}_{2g+2}(\mathbb{D}^2)$ be the configuration of unordered $2g+2$ points in the ...

**12**

votes

**0**answers

132 views

### Which spherical space forms embed in $S^4$?

Is there any hope of getting a classification of which 3-dimensional spherical space forms are smoothly embeddable in $S^4$? I read that lens spaces cannot embed in $S^4$, but some other spherical ...

**8**

votes

**1**answer

178 views

### Essential Klein bottle in simply connected symplectic 4 manifolds

Consider the following question:
Let $X$ be a simply connected, symplectic 4-manifold. Does there exists a smoothly embedded Klein bottle $K\subset X$ such that the following conditions are both ...

**8**

votes

**1**answer

276 views

### Stiefel-Whitney class of fibre bundles

Let $F,E,B$ be manifolds (we may assume them to be compact, without boundary if necessary) and $$F\to E\to B$$ be a fibre bundle. Suppose $$w(F), w(B),$$ the Stiefel-Whitney class of $F$ and $B$, are ...

**5**

votes

**0**answers

93 views

### configuration space of Riemannian manifolds with a parameter on the distance of distinct points

Let $M$ be a Riemannian manifold. For any $\epsilon\geq 0$, we define the $k$-th ($k=1,2,\cdots$) "$\epsilon$-configuration space" as
$$
F(M,k,\epsilon)=\{(x_1,\cdots,x_k)\in M^k\mid d(x_i, x_j)>\...

**7**

votes

**0**answers

166 views

### Translation of Haken's paper “Theorie der Normalflächen”

Haken's paper "Theorie der Normalflächen" is one of the formative papers in the development of normal surface theory and provides an algorithm for detecting the unknot.
While there are now a variety ...

**8**

votes

**0**answers

171 views

### Geometric argument for “easy” part of Jordan-Brouwer separation theorem without local flatness

Let $M^n \subset \mathbb{R}^{n+1}$ be an $n$-dimensional compact connected topological submanifold. The Jordan-Brouwer separation theorem says that $\mathbb{R}^{n+1} \setminus M^n$ contains two ...

**23**

votes

**2**answers

671 views

### SnapPea for the uninitiated

SnapPea (http://www.math.uic.edu/~t3m/SnapPy/) is a program with extensive facilities for doing various kinds of calculations with hyperbolic 3-manifolds. The official documentation assumes that the ...

**4**

votes

**1**answer

113 views

### Symmetry of a distance metric for a generating set of Topology

I was trying to prove that $\epsilon$-balls defined based on the shortest travel-time distance in a transportation network is a valid generating set for a topology of points on a transportation ...

**3**

votes

**1**answer

274 views

### Connections to physics, geometry, geometric probability theory of Euler's beta integral (function)

Euler"s integral for the beta function $B(s,\alpha) = $ (with $x = 1$)
$$ \frac{(s-1)!(\alpha-1)!}{(s+\alpha-1)!} x^{s+\alpha-1} = \int_0^\infty t^{s-1}\; H(x-t) \; (x-t)^{\alpha-1} dt = \int_0^x t^{...

**15**

votes

**2**answers

561 views

### formula for Eta invariant

Hirzebruch's signature formula is not valid for manifolds with boundary.
An error term is introduced by Atiyah-Patodi-Singer to fix it.More precisely:
$$sign (M)=L(M)[M]+\eta(\partial M)$$
Yet ...

**5**

votes

**0**answers

94 views

### Sharp isoperimetry in the discrete Heisenberg group

The exact shape of the set which has the best isoperimetry in the continuous Heisenberg is (from what I know) a difficult open problem. This brought to wonder what is known in the discrete case?
More ...

**2**

votes

**1**answer

77 views

### distinct multiple points in a space with at least one point lying in a subspace

Let $X$ be a topological space and $A$ a subspace of $X$. Given $k\geq 2$, let the unordered configuration space be
$$
B(X,k)=\{(x_1,x_2,\cdots,x_k)\in X^k\mid x_i\neq x_j \text{ for any } i\neq j\}
$$...

**6**

votes

**1**answer

205 views

### Can one twist fibred knots and still get fibred knots?

Suppose we have a fibred knot $K$ with a fiber surface $F$ and let $c$ be an unknot disjoint from $F$ (but not homotopically trivial in the complement of $F$). Is it possible that every twist along $c$...

**6**

votes

**0**answers

393 views

### Prerequisites for reading Gregory Perelman's work

What are the prerequisites for understanding the work of Perelman concerning the Poincaré conjecture?
I am referring to the last three papers here.

**5**

votes

**0**answers

93 views

### Heegaard diagrams of prime 3-manifolds

Are there some known results which give a classification of closed prime 3-manifolds up to their Heegaard diagrams? (That is, providing a collection of Heegaard diagrams which exhausts all prime ...

**9**

votes

**1**answer

223 views

### Diameter of hyperbolic 3-manifolds

Is there a good method to estimate the diameter of a closed hyperbolic 3-manifold?
I am particularly interested in know the diameter of the Weeks manifold.

**13**

votes

**1**answer

319 views

### Fundamental group of the space of immersions of the 2-sphere in 3-space modulo diffeomorphisms of the first

In a previous Mathoverflow question, we saw that the fundamental group of the space $Imm(S^2,\mathbb{R}^3)$ of immersions the 2-sphere in ordinary 3-space is isomorphic to $\mathbb{Z}/2 \times \mathbb{...

**11**

votes

**1**answer

221 views

### Homotopies with prescribed regular values

Let $M_1$ and $M_2$ be connected smooth manifolds and let $f_0,f_1:M_1 \rightarrow M_2$ be homotopic smooth maps such that some fixed point $p \in M_2$ is a regular value for both $f_0$ and $f_1$. ...

**0**

votes

**1**answer

111 views

### cohomology ring of the fundamental group of unordered configuration space

From the lecture notes INTRODUCTION TO CONFIGURATION SPACES AND THEIR
APPLICATIONS, p. 18, I find:
Os it possible to derive the cohomology ring $H^*(Conf(S,k)/\Sigma_k;\mathbb{Z}_2)$ from the above ...

**2**

votes

**0**answers

52 views

### Stiefel manifolds and “simplicial complex chromated Sitefel manifolds”

Let $K$ be a simplicial complex whose vertices are labelled by $1,2,\cdots,k$. I want to define a variant concept of the open Stiefel manifolds
$$
V_K(\mathbb{R}^n):=\{(v_1,v_2,\cdots,v_k)\in\prod_k\...

**29**

votes

**1**answer

596 views

### Classifiying sphere eversions

For a year I have been giving lectures on a (probalby) new way to present an explicit sphere eversion. These lectures include a review of many other explicit eversions that have been described, as ...