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
109 views

Existence of non-negative extensions of smooth functions on axes

I am struggling to solve an extension problem of smooth functions, and I would like someone to help me. The setting is as follows: Let $X_1$, $X_2$, and $T$ be either the real lines $\mathbb R$ or ...
6
votes
1answer
275 views

Construction of invariants of 4-manifolds with the Kirby calculus

I'm an undergraduate student, interested in the low dimensional topology, in particular, the 4-manifold theory. I have a question. In the knot theory, the Reidemeister moves play fundamental roles. ...
2
votes
0answers
168 views

Extra large spherical joins

If $X$ and $Y$ are piecewise spherical complexes, then their spherical join $X * Y$ is CAT(1) if and only of $X$ and $Y$ are CAT(1) (see the appendix of the first Charney-Davis paper below). One of ...
0
votes
1answer
169 views

group actions on fibre bundles

Suppose that we have a group $G$ acting on the spaces $E$ and $B$. Suppose moreover that we have fibre bundles $\xi$ and $\eta$ in the following commutative diagram If $\xi$ is a trivial bundle, ...
3
votes
0answers
116 views

Uniform continuity of length function on geodesic currents

I'm starting to study geodesic currents and I have a question concerning uniform continuity. Let's take $S$ a closed surface of genus $g$ and $GC(S)$ the space of geodesic currents on $S$ (as it is ...
10
votes
1answer
335 views

Wild half-line in a Euclidean space

Is there an $m$-dimensional simplicial complex $S$ with the following properties: The cone over $S$ is homeomorphic to $\mathbb{E}^{m+1}$. Here $\mathbb{E}^{m+1}$ denoes the $(m+1)$-dimensional ...
3
votes
0answers
146 views

Are all finitely generated subgroups of SL2 LERF? [closed]

Let $H \leq \mathrm{SL}_2(\mathbb{R})$ be a finitely generated subgroup. Must $H$ be LERF? A group $H$ is said to be LERF (locally extended residually finite), or subgroup separable, if its ...
1
vote
0answers
93 views

contact structure on double branched covers of $S^3$

We can assign a natural contact structure to double branched cover of a transverse knot $k$ in the $3$-sphere with its standard contact structure $(S^3,\xi_{st})$, as described ...
5
votes
1answer
138 views

lower bound for Perron-Frobenius degree of a Perron number

A Perron number is an algebraic number which is greater than one in absolute value and is greater than all of its Galois conjugates in absolute value as well. Lind's theorem states that any Perron ...
7
votes
1answer
166 views

Subgroups of the mapping class group of a surface generated by Dehn twists

Let $G$ be the mapping class group of a surface of genus $g > 1$. Is it known for which positive integer $k$ one can find a subgroup $H$ of $G$ generated by a finite number of Dehn twists and a ...
8
votes
2answers
222 views

Quasi-isometric rigidity of Nil

Let $Nil$ be the unique simply connected non-abelian three-dimensional nilpotent Lie group, i.e. the group of upper triangular matrices with all the eigenvalues equal to 1 (this group is also known as ...
13
votes
3answers
672 views

Classification of knots by geometrization theorem

I read this interview with Ian Agol, where he says: "...I learned that Thurston’s geometrization theorem allowed a complete and practical classification of knots." My question is: How does ...
6
votes
0answers
89 views

Survey of known results on equivariant transversality

Most basic differential topology theorems carry over to the equivariant case with mild modifications; see for instance Wasserman's paper. One thing that fails (more or less obviously) is equivariant ...
3
votes
1answer
143 views

cohomology module of unit tangent vector bundles over spheres

Let $S^m$ be the $m$-sphere and $\tau (S^m)$ the sphere bundle consisting of unit tangent vectors in the tangent bundle $TS^m$. Then we have a fibration $$ S^{m-1}\longrightarrow ...
6
votes
2answers
508 views

quotient space of Eilenberg-MacLane space

Let $\pi$ be a group and $K(\pi,1)$ the Eilenberg-MacLane space. Let $G$ be a finite group acting on $K(\pi,1)$ such that the following is a covering map $$ K(\pi,1)\longrightarrow K(\pi,1)/G. $$ ...
4
votes
2answers
252 views

homotopy equivalence between configuration spaces

Let $M$ be a $m$-dimensional compact manifold without boundary and $W(M)$ the non-compact $CW$-complex obtained by glueing $[0,1)\times (0,1)^{m-1}$ to $M$, identifying the boundary ...
2
votes
1answer
204 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
270 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
181 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
119 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 \, ...
12
votes
1answer
233 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
108 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
369 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
159 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
90 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
129 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
53 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
127 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
213 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
156 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
69 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
427 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
206 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
285 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
130 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
428 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
308 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?
21
votes
3answers
833 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
743 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
196 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
361 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
139 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
179 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
407 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
252 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 ...