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 ...

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2
votes
1answer
201 views

isotopy equivalence (topological meaning) between $CW$-complexes

Let $M$ and $N$ be $CW$-complexes. Definition. (different from the isotopy notion in geometry of submanifolds). A (topological) isotopy is a fibre-wise continuous map $$ F: M\times ...
9
votes
1answer
269 views

embedding of quaternionic projective spaces

Let $\mathbb{H}P^m$ be the $m$-th quaternionic projective space. What is the smallest integer $N$ such that there exists an embedding $$ \mathbb{H}P^2\longrightarrow \mathbb{R}^N? $$ Are there any ...
7
votes
1answer
179 views

homological 2 dimensional groups

In a Commentarii Mathematici Helvetici paper by Benno Eckman and Heinz Müller in 1980 (volume 50, pages 510-520) proved that poincaré Duality Groups of dimension 2 with positive first ...
5
votes
1answer
118 views

Ascending surfaces in the 4-ball

Let the standard symplectic structure on $B^4$ (viewed in $\mathbb{R}^4$ or $\mathbb{C}^2$) be given by $\omega=(1/2) d \eta$, for \begin{align*} \eta &:= x_1 \, dy_1 - y_1 \, dx_1 + x_2 \, ...
13
votes
1answer
230 views

Which compact 3-manifolds with boundary embed in $\mathbb{S}^3.$

This is a more sensible (IMHO) restatement of this question: Which Compact $3$-manifolds with boundaries embed in $\mathbb{S}^3?$ Is there any hope of a characterization?
1
vote
0answers
107 views

cohomology ring of compact submanifolds of Euclidean spaces

Suppose we have a compact $m$-dimensional submanifold $M$ of $\mathbb{R}^N$ and we want to know the cohomology ring $H^*(M;\mathbb{Z})$. Let $\epsilon>0$ and a $m$-dimensional finite simplicial ...
14
votes
2answers
354 views

Can an oriented closed $n(\geq 2)$-dimensional manifold be embedded in $\mathbb{R^{2n-1}}$

Can anyone provide me an example of an orientable closed manifold $M$ of dimension $n\geq 2$, which cannot be embedded in $\mathbb R^{2n-1}$? I know this is certainly not true when $n=1$, i.e. ...
1
vote
1answer
155 views

torsion part of the cohomology module of configuration spaces of manifolds

Let $M$ be a manifold and $C_n(M)$ the $n$-th unordered configuration space consisting of unordered $n$-tuples of distinct points in $M$. The mod $p$ homology module, $p$ prime, and the rational ...
3
votes
1answer
89 views

self-Whitney sum of the canonical vector bundle on Grassmannians

Let $G_{k}(\mathbb{R}^N)$ be the Grassmannian manifold consisting of $k$-subspaces in $\mathbb{R}^N$. There is a canonical $k$-dimensional vector bundle $$ \gamma_{k,N}: \mathbb{R}^k\longrightarrow ...
3
votes
0answers
126 views

Finiteness for 2-dimensional contractible complexes

While thinking about graph-complex and related operadic stuff, I found a quite interesting (at least for me) question. However, I'm a novice in the algebraic topology, so I'm unable to resolve it by ...
2
votes
1answer
52 views

Complexity of recognizing equivalent translation surfaces

"A translation surface is a union of polygons with pairs of parallel edges identified by translation, up to cut and paste equivalence." I take that succinct (and not fully precise) definition ...
3
votes
1answer
125 views

Constructing a “nice” cobordism

Denote by $\Sigma_g$ the closed, orientable surface of genus $g$. I want to construct a cobordism $M_g$ between $\Sigma_g$ and $\Sigma_{g+1}$ with the following two nice properties: 1) $M_g$ is an ...
5
votes
2answers
210 views

Poisson structures on non-smooth manifolds with singularities

It's very known how we can describe a Poisson structure on a manifold $M$, where $M$ is a smooth manifold, but what about a Non-smooth manifold with singularities? In section $(2)$ of the paper The ...
4
votes
1answer
152 views

contact surgery diagram on Brieskorn manifolds

For the Brieskorn manifold $\Sigma(p,q,r)=\{z_1^p+z_2^q+z_3^r=0\} \cap S^5 \subset C^3$, replacing zero with $\epsilon$ in the above, realizes $\Sigma$ as boundary of a Stein domain which induces a ...
2
votes
1answer
115 views

Under which conditions is it possible to find points with same distances under bi-Lipschitz map [closed]

Given two metric spaces $(X,d_X), (Y, d_Y)$, a bi-Lipschitz map $f:X \to Y$ and a finite set of points $\{x_1, \ldots, x_n\} \in X$. Consider in addition, that $X$ is a vector space over $\mathbb{R}$, ...
0
votes
0answers
68 views

Rigidity of lower-dimensional lattices in Euclidean groups

Informal intro / motivation: Suppose I have an infinite set of atoms arranged in a 2D periodic crystalline "sheet". By crystalline I simply mean that it is preserved by the action of integer linear ...
24
votes
2answers
535 views

Can one deform an immersion of a 3-manifold in $\mathbb R^4$ to an embedding in $\mathbb R^6$?

Let $M^3$ be an oriented 3-manifold, and let $f:M^3\looparrowright \mathbb R^4$ be a codimension one immersion. Is it possible to find a small deformation of the composite map $$ M^3 \to \mathbb R^4 ...
7
votes
1answer
421 views

A conjecture of Cheeger about intersection cohomology and $L^2$- cohomology

Let $X$ be a projective variety and let $D$ be a simple normal crossings divisor on $X$ Does $$IH^*(X;\mathbb C)\cong H_{(2)}^*(X\setminus D;\mathbb C)$$ hold true for each Kähler metric ...
19
votes
0answers
212 views

“High-concept” explanation for proof of a theorem of Ochanine?

See Akhil Mathew's notes on Ochanine's theorem for elliptic genera here and here. Let $\phi: \Omega_{SO} \to \Lambda$ be a genus. We might ask when $\phi$ satisfies the following multiplicative ...
4
votes
2answers
202 views

Is the following 3-manifold always a trivial I-bundle over a surface?

Let $M$ be a compact, orientable and irreducible 3-manifold with with boundary consisting of two incompressible components $N_0,N_1$, with $N_i \stackrel{f_i}{\cong} S_g$ for some diffeomorphism ...
13
votes
1answer
284 views

Exotic line arrangements

I would like to discuss the following problem. Hopefully, you will suggest to me some ideas and bibliography. At first I will provide some basic definitions to set up the notation. Let us consider ...
3
votes
2answers
126 views

f vectors of simplicial complexes homeomorphic to n dimensional spheres

In dimension 2, the euler poincare formula restricts the incidence properties of edges in a triangulation of a surface. Are there analogous generalizations for higher dimensions, like elaborations ...
4
votes
2answers
423 views

Thurston geometries---the geometry of the universal cover of $SL(2, \mathbb{R})$

In one of the eight Thurston geometries there is the geometry of the universal cover of $SL(2, \mathbb{R})$. But from the algebraic point of view $PSL(2,\mathbb{R})$ is sufficient for building ...
6
votes
1answer
285 views

Parameterization of a knotted surface?

I'm looking for a parameterization $(x_1(u,v),x_2(u,v),x_3(u,v),x_4(u,v))$ of a knotted sphere in $\mathbb R^4$. How might one go about finding such a parameterization?
20
votes
3answers
822 views

What are the implications of the simple loop conjecture?

Drew Zemke recently posted a preprint on arXiv proving the Simple Loop Conjecture for 3-manifolds modeled on Sol. Simple Loop Conjecture: Consider a 2-sided immersion $F\colon\, \Sigma\rightarrow ...
11
votes
4answers
739 views

Distance between two knots

Are there some well-studied functions defining natural distance measures between two knots? One can imagine a function that counts, say, the minimum number of moves, each of which passes one strand of ...
4
votes
1answer
192 views

Compact open topology on the space of geodesics

I'm new in the field, so I'm sorry in advance if my question is too naive. Let's consider $S$ a surface of genus $g\ge 2$ with an hyperbolic metric $g$. Let's call $\mathcal{S}(S)$ the set of closed ...
5
votes
1answer
225 views

abelian and nonabelian parts of Aut($\widehat{F_2}$)

Let $F$ be the free profinite group on two generators. Let $\text{IA}(F) := \ker\left(\text{Aut}(F)\rightarrow GL_2(\widehat{\mathbb{Z}})\right)$, the group of "IA automorphisms" of $F$. (I'm also ...
2
votes
1answer
353 views

embeddings of product of spheres in Euclidean spaces [closed]

I notice that $T^2=S^1\times S^1$ can be embedded in $\mathbb{R}^3$ as a hypersurface (submnaifolds of codimension 1). In general, (1). could the product of spheres $S^{m_1}\times\cdots\times ...
0
votes
0answers
50 views

Volume growth of balls II

Let $b:(0,\infty)\to (0,\infty)$ be monotonically increasing. Call $b$ limit-tight, if $$ \lim_{\varepsilon\to 0}\ \limsup_{T\to\infty}\frac{b(T-\varepsilon)}{b(T)} =\lim_{\varepsilon\to 0}\ ...
4
votes
1answer
138 views

covering map from spheres to projective spaces and the associated vector bundle

Let $S^n$ be the $n$-sphere and consider a $2$-sheeted covering $$ S^n\longrightarrow\mathbb{R}P^n. $$ We have an associated vector bundle $$ \xi: \mathbb{R}^2\longrightarrow ...
3
votes
1answer
177 views

An inequality for Fuchsian groups?

Let $G$ be a finitely generated Fuchsian group. (i.e. a discrete subgroup of $\mathrm{PSL}_2(\mathbb{R})$). Is it true that $d(G) < 2\beta_{2}^1(G) + 1$ ? Here, $\beta_{2}^1(G)$ stands ...
5
votes
3answers
398 views

classifying space of orthogonal groups

Let $O(n)$ be the $n$-th orthogonal group and $O$ be the direct limit of $O(n)$ with respect to $n$. Let $BO(n)$ and $BO$ be the classifying spaces. Question: Why $BO$ is an $H$-space? My supervisor ...
3
votes
2answers
248 views

rational cohomology of symmetric groups

Let $\Sigma_k$ be the $k$-th symmetric group and $B\Sigma_k$ be its classifying space. How to prove: for any $n\geq 1$ and the $n$-skeleton $sk_n (B\Sigma_k)$, there exists a finite dimensional ...
4
votes
1answer
77 views

Example of a doubly degenerate surface group not coming from a pseudo-Anosov mapping torus

Doubly degenerate surface groups are discrete subgroups of $PSL(2,\mathbb{C})$ whose limit set is all of $S^2$, the boundary of $\mathbb{H}^3$. A standard example of such a group is given as follows: ...
3
votes
1answer
193 views

“Ambient homotopy” between preimages under a fiber bundle?

Choose a notion of an "ambient homotopy" between maps of topological spaces. For example, say that two embeddings $Y \rightarrow X$ are ambiently homotopic if there is a path between them in the space ...
4
votes
1answer
146 views

Which 3-manifolds have positive rank gradient?

For which $3$-manifolds $M$ is the fundamental group $\pi_1(M)$ finitely generated and has positive rank gradient? Recall that the rank gradient of a finitely generated group $G$ is defined to ...
2
votes
0answers
95 views

Chern character of finite $CW$-complexes and rational Pontrjagin class of vector bundles

Let $K$ be a finite $CW$-complex. Could you give any references or explanations for the following two items? I do not understand. Thanks! (1). The Chern character from $\tilde{KO}^0(K)$ to the ...
2
votes
1answer
128 views

Engulfing Kleinian groups?

Let $G$ be a Kleinian group, and let $H \lneq G$ be a finitely generated subgroup. Must there be a proper finite index subgroup $U$ of $G$ containing $H$ ? I know that this is true for Fuchsian ...
1
vote
1answer
92 views

Lamination as limit of arcs

I am reading Bonahon's notes on closed curves, in particular the part about hyperbolic laminations. In his notes Bonahon illustrates some examples as why laminations should be "limit curves" on ...
5
votes
0answers
101 views

Second homology of mapping class group via universal coefficient theorem

Let $S_g$ be the closed, orientable surface of genus $g\ge 2$ and $Mod(S_{g,1})$ the mapping class group of homeomorphisms fixing a base point $x_0\in S_g$ modulo isotopies fixing $x_0$. It is known ...
5
votes
1answer
210 views

how to prove the $n$-times self-product of a map is null-homotopic

Let $k$ be a fixed positive integer and $\Sigma_k$ the $k$-th symmetric group. By letting $\Sigma_k$ permuting an orthonormal basis of a $k$-dimensional Euclidean space, there is a "regular ...
8
votes
0answers
168 views

Which map realizes the isomorphism $KO_n(X)\otimes \mathbb{Q}\to \bigoplus_{i\in\mathbb{Z}}H_{n-4i}(X;\mathbb{Q})$?

The description of the real $KO$-homology groups $KO_n(X)$ can be given abstractly as maps to the real K-theory spectrum $KO$ smash $X$, or via triples $(M,x,\phi)$ where $M$ is a closed manifold, ...
6
votes
0answers
389 views

Smoothing a piecewise smooth manifold

Let $M \subset \mathbb{R}^d$ be a piecewise smooth $2$-manifold. Let $C$ be a polyhedral complex that covers $\mathbb{R}^d$ and contains faces of dimension $[0,d]$. Since $M$ is a $2$-manifold, we can ...
6
votes
2answers
213 views

How bad is the modular space?

I'm wondering if there is some results about the quotient space $\mathbb{H}^{3}/PSL(2,\mathcal{O}_{K})$? Do we know something about its homology or homotopy groups ? $\mathbb{H}^{3}$ is the hyperbolic ...
2
votes
0answers
86 views

order of elements in a mapping space

Let $B$ be a finite CW-complex and $\xi$ be a vector bundle over $B$ with structure group $\Sigma_n$, the $n$-th symmetric group. Then corresponding to $\xi$, we have a classifying map $$ g\in \tilde ...
3
votes
2answers
132 views

equivariant embeddings from the k-th configuration space to the k+1-th configuration space

Let $S$ be a closed, orientable surface in $\mathbb{R}^3$ and $S'$ be the manifold $S\setminus\text{one point}$. Let $F(S',k)$ be the $k$-th (ordered) configuration space on $S'$. It is claimed in ...
5
votes
0answers
70 views

How to obtain a faithful representation of $\pi_{1}(\Sigma_{2})$ into ${\rm PSL}(2,\mathbb{Z}[i])$?

Does there exist such a representation? In the title, $\Sigma_{2}$ means the closed orientable surface of genus 2. I once heard of this or something like it, but not quite sure. Thanks to everyone!
10
votes
3answers
371 views

When a compact topological manifold with boundary is a ball?

Let $X$ be a compact topological manifold with boundary. Suppose its interior is homeomorphic to $\mathbb{R}^n$. Is $X$ homeomorphic to a (closed) ball? Context: I want to show that a certain ...
6
votes
2answers
316 views

3-manifolds homotopy equivalent to a surface

I have heard that an open, orientable 3-manifold $X$ (non-compact, without-boundary) that is homotopy equivalent to an orientable surface $S_g$ must itself already be homemorphic to $S_g \times ...