Topology of cell complexes and manifolds, classification of manifolds (e.g. smoothing, surgery), low dimensional topology (e.g. knot theory, invariants of 4-manifolds), embedding theory, combinatorial and PL topology, geometric group theory, infinite dimensional topology (e.g. Hilbert cube manifolds,...

learn more… | top users | synonyms (1)

3
votes
0answers
128 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
1answer
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
1answer
254 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
1answer
188 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
6answers
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
0answers
94 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
2answers
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
1answer
486 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
1answer
160 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
1answer
489 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
0answers
205 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
1answer
254 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
1answer
196 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
0answers
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
1answer
174 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
1answer
271 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
0answers
92 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
0answers
164 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
0answers
169 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 ...
22
votes
2answers
666 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
1answer
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
1answer
270 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
2answers
558 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
0answers
93 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
1answer
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
1answer
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
0answers
388 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
0answers
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
1answer
220 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
1answer
316 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
1answer
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
1answer
109 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
0answers
51 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
1answer
589 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 ...
2
votes
0answers
142 views

A question of braid words

Let $(W,S)$ be a Coxeter group, let $B(W,S)$ be the corresponding braid or Artin-Tits group. Set $S=\{s_1,\dots, s_n\}$ and denote by $\bf{S}=\{\sigma_1,\dots, \sigma_n\}$ the corresponding generators ...
1
vote
0answers
41 views

Thomsen Blaschke condition

I am reading a paper (Paper 1: https://ideas.repec.org/p/cwl/cwldpp/76.html, that cites another paper ( Paper 2) for its proof. Paper 1, page 1, line 10 says : Consider the topological image G of a 2-...
3
votes
1answer
503 views

characteristic classes of homotopy equivalent manifolds

Let $M,N$ be two manifolds of different dimensions. Suppose $M\simeq N$, i.e. $M$ is homotopy equivalent to $N$. Do the Stiefel-Whitney classes of the tangent bundles of $M$ and $N$ equal $$ w(TM)=w(...
3
votes
1answer
133 views

cohomology ring of infinite iterated loop space

What is the cohomology ring $$ H^*(\Omega^\infty \Sigma^\infty (S^m\vee S^n);\mathbb{Z}_2)? $$ I already write out the graded-vector-space basis using Dyer-Lashof operations, but I do not know how to ...
4
votes
1answer
179 views

Current status of the linearity of mapping class group

In the paper A faithful linear-categorical action of the mapping class group of a surface with boundary it is claimed that it was (as of 2014) still unknown that mapping class group is linear, but the ...
3
votes
1answer
106 views

unordered configuration space over spheres and Euclidean spaces

For a topological space $X$, let $B(X,k)$ be the $k$-th unordered configuration space. Then $$ B(\mathbb{R}^n,2)\simeq \mathbb{R}P^{n-1}, $$ $$ B(S^n,2)\simeq \mathbb{R}P^n. $$ Hence $ (*) $ $$ B(\...
5
votes
1answer
197 views

Are all transversely oriented, transversely measured foliations given by closed forms?

This paper, "AF-algebras and topology of 3-manifolds" seems to claim on page 5 that a [transversely] oriented measured foliation on a compact surface is given by a closed form. On the other hand, the ...
6
votes
2answers
338 views

Is there a nonabelian free group inside a group of positive rank gradient?

Let $G$ be a finitely generated residually finite group with positive rank gradient, and let $F_2$ be the free group on $2$ elements. Must there be an embedding $i \colon F_2 \to G$ ? A group $G$...
6
votes
3answers
184 views

Probability of random geodesics on the half-sphere intersecting

4 end points (a,b,c,d say) are chosen uniformly randomly and connected a to b and c to d by two geodesics on the 2-dim half-sphere. Here, uniform means that, probability that a point lies on a surface ...
4
votes
1answer
179 views

Anosov representations and boundaries of (harmonic) maps

Let $\Sigma_g$ be a closed hyperbolic surface and $\rho\colon\pi_1\Sigma_g\to G$ an Anosov representation into a suitable Lie group. By definition of Anosovness, one has a $\rho$-equivariant ...
9
votes
2answers
345 views

For a 3-manifold $Y$, when does $Y\times S^{1}$ admits a Riemannian metric with positive scalar curvature?

Let $Y$ be an orientable, smooth 3-manifold and let $X=Y\times S^{1}$. My question is that: when does $X$ admits a Riemannian metric with positive scalar curvature? An obvious case is when $Y$ ...
2
votes
0answers
50 views

When do positively invariant subset contain a given set?

Non-triviality of the Conley index for an isolating neighborhood $N$ and a flow $\varphi$ can be used to prove non-emptyness of the related isolated invariant set. In particular, if $N$ doesn't ...
4
votes
1answer
127 views

Why almost every geodesic arc is generic?

In his paper "Geodesic laminations on surfaces", Bonahon gave the definition of generic arc and a property as following. An arc $k$ is generic (with respect to simple geodesics) if it is transverse ...
3
votes
1answer
114 views

Existence of tubular neighborohoods of locally flat topological embeddings

Suppose $X$ is a topological manifold and $Y \subset X$ is a locally flat submanifold. We know that $Y$ doesn't necessarily have a tubular neighborhood. My definition of a tubular neighborhood of $Y$ ...
10
votes
0answers
183 views

Do homology classes have “special” representatives?

Recall that, according to Hodge, de-Rham cohomology classes of "nice enough" manifolds have "special" representatives - namely, harmonic forms. Now, how does one choose a "special" one among ...
6
votes
2answers
267 views

Fibered example of topologically slice knots

Is there any known example of fibered knot which is topologically slice but not (expected to be) smoothly slice?