**16**

votes

**0**answers

192 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

**2**answers

193 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 ...

**6**

votes

**1**answer

175 views

### Exotic line arrangements

I would like to discuss about the following problem. Hopefully, you will suggest me some ideas and bibliography.
At first I provide some basic definitions to set up the notation.
Let us consider a ...

**3**

votes

**2**answers

120 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 ...

**3**

votes

**0**answers

69 views

### Is the doubling dimension invariant under Möbius maps?

Is the doubling dimension of a metric space $(X,d)$ invariant under maps $f:(X,d) \to (X,d')$, which are the identity on $X$ and which map the metric $d$ to a metric $d'$ which has the same cross ...

**4**

votes

**2**answers

415 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

**1**answer

276 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?

**18**

votes

**3**answers

775 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

**4**answers

709 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

**1**answer

185 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

**1**answer

223 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

**1**answer

331 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

**0**answers

48 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

**1**answer

132 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

**1**answer

174 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 ...

**4**

votes

**3**answers

355 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

**2**answers

244 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

**1**answer

75 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

**1**answer

189 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 ...

**1**

vote

**0**answers

88 views

### Higher connectedness of Rips complexes

For $G$ a countable discrete group, if it is finitely generated resp. finitely presented, we know that there is $R_0$ resp. $R_1$ such that for every $R\geq R_i$ the Rips complex ...

**4**

votes

**1**answer

140 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

**0**answers

94 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 ...

**1**

vote

**1**answer

117 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

**1**answer

89 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

**0**answers

96 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

**1**answer

205 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

**0**answers

156 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

**0**answers

307 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

**2**answers

205 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

**0**answers

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

**2**answers

126 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

**0**answers

67 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

**3**answers

359 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 ...

**5**

votes

**2**answers

294 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 ...

**3**

votes

**1**answer

201 views

### the “Kahn-Priddy map” and “multiplicative $p$-local equivalence”

The following is a part of a paper that I need to understand
I totally do not know the argument. Could you explain? Thanks.
Let $\Sigma_n$ be the $n$-th symmetric group and $\Sigma_\infty$ be the ...

**0**

votes

**0**answers

91 views

### Does anyone know any applications of CW-complexes in graph theory?

As everyone knows :P, a graph is a CW-complex of dimension 1. Knowing that, are there any interesting results in graph theory that arise from working with CW-complexes? And more specifically, in ...

**0**

votes

**0**answers

118 views

### when is “fibering” preserved under homotopy equivalence

Suppose I have an oriented $F$ bundle over $B$ with total space $E$ (all of the three are closed manifolds) and i have a closed manifold $E'$ which is homotopy equivalent to $E$.Is there any condition ...

**1**

vote

**0**answers

84 views

### Homology of spherical braid groups

By the spherical braid group, I mean the fundamental group of the configuration space of distinct unordered points in $S^2$. I am wondering what is known about the group homology of the spherical ...

**4**

votes

**1**answer

133 views

### Homology 3-sphere with a unique Stein-fillable contact structure

Are there any known examples of oriented integer homology 3-spheres $Y$ (besides $S^3$) which have exactly one Stein-fillable contact structure up to isotopy? Failing that, what are the known examples ...

**1**

vote

**3**answers

185 views

### reference on complex dynamics

Please someone suggest me some reference on the topic "Complex Dynamics". I want a brief geometric treatment from the root level. I have graduate level background on complex analysis, riemannian ...

**2**

votes

**2**answers

238 views

### Is the following 3-manifold irreducible?

We start with product manifold $ X := S_g \times (0,1)$, where $S_g$ is a closed, orientable surface of genus $g \geq 1$. Since this has $\mathbb R^3$ as its universal cover, $X$ is irreducible. Now ...

**6**

votes

**1**answer

159 views

### homotopy groups of an orbifold

The isometry group of the 3-dimensional hyperbolic space $\mathbb{H}^{3}$ is $PSL(2,\mathbf{C})$. What are the homotopy groups of the quotient space $\mathbb{H}^{3}/PSL(2,\mathbf{Z})$ ?

**7**

votes

**3**answers

155 views

### Cyclic groups acting on balls, and interior fixed points

Let a finite cyclic group $G = \mathbb Z/n$ act continuously on an open $d$-ball $B^d$. Suppose further that this action extends to the closed ball $\overline{B^d}$. Is there necessarily a fixed point ...

**3**

votes

**1**answer

182 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×I has a collapsible subdivision.
What is the status of this conjecture ...

**3**

votes

**0**answers

77 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

96 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

120 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

116 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

187 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^*: ...

**5**

votes

**0**answers

108 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 ...