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
0answers
146 views

What invariants are of great concern in the field of 3-manifolds and why? How much do we know about them? [closed]

I am curious about 3-manifolds though I know little. Here I am trying to know what invariants people in this field are interested in. The following are what I have known and what I particularly want ...
0
votes
3answers
318 views

Good books on Geometric Theory of Dynamical Systems

I am looking for a good book on Geometric Theory of Dynamical Systems . I found Geometric Theory of Dynamical Systems by Jr. Palis myself,but it's very old, anyway i would like to find a pure ...
6
votes
0answers
121 views

Is there an analogue of Mostow-Palais equivariant embedding theorem for noncompact groups

Let $M$ be a (Hausdorff) smooth compact manifold and $G$ a Lie group acting smoothly on $M$. If $G$ is compact then, by Mostow-Palais theorem, there exists an equivariant smooth embedding $M\to ...
6
votes
1answer
207 views

Is there “nonorientable Heegaard Floer homology”?

I have a Heegaard diagram which produces a non-orientable 3-manifold. I want to know any 3-manifold invariant which can be calculated from Heegaard diagrams for non-orientable 3-manifold. (As far as I ...
15
votes
2answers
464 views

Distinct manifolds with the same configuration spaces?

For a space $X$, let $C_k X$ denote the space of configurations of $k$ distinct unordered points in $X$. What is an example of a pair of smooth manifolds $M$ and $N$ that are not homeomorphic but ...
2
votes
1answer
116 views

Density of ends of long words in a hyperbolic group

Let $G$ be a Gromov hyperbolic group with a generating set $S$. For each $g \in G$, let $\xi_g$ be the point in the ideal boundary corresponding to the sequence $(g^n)$. Let $l_S(g)$ be the word ...
3
votes
1answer
133 views

Link surgery on $S^2\times S^1$

Given $n$ points $p_1,\dots,p_n$ in $S^2$ one gets a product link $L_n=\{p_1,\dots,p_n\}\times S^1$ inside the closed 3-manifold $S^2\times S^1$, which can be looked at as a trivially framed link (by ...
1
vote
0answers
42 views

Natural isomorphism between locally finite homology and homology of one-point compactifcation of a forward tame ANR

Let $X,Y$ be a locally compact, separable, metric ANR that is forward tame, which means that for some closed subset $V\subseteq X$ such that $\overline{X\smallsetminus V}$ is compact a proper map ...
6
votes
1answer
331 views

consequence of Novikov conjecture

Novikov conjeture is a famous open problem in Geometric topology.It predicts that higher signature is oriented-homotopy invariant. http://en.wikipedia.org/wiki/Novikov_conjecture I am a student ...
5
votes
0answers
124 views

Uniform incentre of collection of quasi convex subspaces in hyperbolic spaces

I'm reading a paper of Wise on cubulations and the following fact is used: Let $H$ be a quasi-convex subgroup of a $\delta$-hyperbolic group and let $H_i, i\in I$ be a finite family of translates of ...
10
votes
1answer
205 views

Why do convex polytope options constrict with dimension, rather than expand?

There are an infinite number of regular polygons in the plane, five regular polyhedra, six regular polytopes in $\mathbb{R}^4$, and then three regular polytopes in every dimension $d > 4$. There ...
8
votes
1answer
148 views

Mapping class groups of small Seifert-fibred 3-manifolds

Are computations of the mapping class groups of small Seifert-fibred 3-manifolds recorded in some convenient location? For most Seifert manifolds working out the mapping class group is easy-enough ...
11
votes
1answer
373 views

Does Gromov's Waist Inequality imply Borsuk-Ulam?

I'm curious if anyone can see a route to get the Borsuk-Ulam theorem from Gromov's waist inequality. For the sake of notation, here's the inequality: Let $S^n$ denote the round unit sphere in ...
2
votes
1answer
68 views

Some questions on partial pseudo anosov maps

Let $S$ be an orientable closed 2-surface with genus at least 2 and $C$ be a non-separating essential simple closed curve in $S$. Denote $S_{C}=S-N(C)$. Let $f$ be a pseudo anosov map of $S_{C}$. ...
4
votes
2answers
166 views

Getting surgery link from Heegaard splitting

From Lickorish-Wallace theorem, every 3-manifold is an integral surgery on a link in $S^3$. From its proof from Saveliev's book, it seems obvious that if I know the Heegaard splitting of a closed ...
1
vote
1answer
79 views

Centralizer of a pseudo-Anosov element

What is the centralizer of a pseudo-Anosov element in the mapping class group of an orientable punctured surface? Is it cyclic? If so, where can I find a proof?
10
votes
1answer
244 views

Mapping class group and CAT(0) spaces

I hope the questions are not too vague. Is the mapping class group of an orientable punctured surface $CAT(0)$ ? Is any of the remarkable simplicial complexes (curve complex, arc complex...) built ...
1
vote
0answers
137 views

The image of homomorphism of fundamental group of closed surface [closed]

$\phi: \pi_1(S)\to \pi_1(S)$ is a homomorphism of fundamental group of closed orientable surface $S$ of genus >=2. If $\phi$ is not an epimorphism, can we find a non-surjective self map $f: S\to S$ ...
5
votes
1answer
191 views

Can an open manifold with positive Ricci curvature be non simply connected at infinity?

The question is in the title, I haven't been able to locate a discussion of these kind of properties.
7
votes
2answers
222 views

How do I efficiently find a sequence of Reidemeister moves between equivalent link diagrams?

In knot theory, two link diagrams are equivalent if and only if they can be related by performing a finite number of Reidemeister moves. But sometimes it is so confusing that I don't know which type ...
11
votes
3answers
247 views

Are there quanitative versions of Thurston's geometrization for manifolds which fiber over $S^1$?

The geometrization theorem tells us: Theorem (Thurston) The mapping torus $M_\phi$ of a pseudo-Anosov diffeomorphism $\phi: S_g \rightarrow S_g$ from a genus $g$ surface to itself admits a ...
6
votes
7answers
898 views

Homotopy groups other than $\pi_1$ : what are they good for? [duplicate]

Homotopy groups $\pi_k$ were introduced before the homology groups $H_k$ in the 1st topology book I read and the 1st topology course I took. Later on $\pi_1$, $H_k$, and $H^k$ appeared in numerous ...
1
vote
1answer
195 views

Sufficient condition for coverings between non-orientable surfaces

Let $X_k$ be the connected sum of $k$ projective planes. I am interested in necessary and sufficient conditions for the existence of a covering $\pi: X_{k'} \to X_k$, where $k$ and $k'$ are integers. ...
3
votes
0answers
204 views

Looking for a necessary and sufficient condition for the polarization $\mathbb{P}$ being positive

My question is about positivity of polarization in Geometric quantization theory. Let $\mathbb{P}$, be a complex polarization on symplectic manifold $(M,\Omega)$. For every $m\in M$, we can define a ...
7
votes
3answers
790 views

Is the list of “known” 3D compact manifolds complete?

"it is an open question if the known compact manifolds in 3-D are complete." This is a quote from Eric Weisstein's CRC Concise Encyclopedia of Mathematics, Second Edition. 2010, p.480. (Google ...
2
votes
1answer
338 views

looking for an identity for higher jet bundle $J^kM$?

We know this fact that the first jet Bundle $J^1M$ is diffeomorphic with $T^*M×\mathbb{R}$.i.e, ($J^1M=T^*M×\mathbb{R}$) Is there something like this identity for higher jet bundle $J^kM$? I editted ...
9
votes
1answer
189 views

Heegaard genus of the hyperbolic dodecahedral space (is it 3 or 4?)

I have a question on the hyperbolic dodecahedral space, first described by C.Weber and H. Seifert in 1933 [Die beiden Dodekaederr\"aume, Math Z. 37 (1933), 237-253]. Is it known whether it admits a ...
3
votes
0answers
132 views

Mapping One Curve to another using Dehn Twists

Let $M$ be an orientable surface with genus $g>1$. Let $\alpha$ and $\beta$ represent two different isotopy classes of essential curves on the surface. Is anyone aware of a technique or algorithm ...
6
votes
0answers
244 views

Space of embeddings of an $n$-ball into an $n$-manifold

Let $M$ be a smooth $n$-manifold without boundary, and let $B$ be the open unit ball in $\mathbb{R}^n$. I am trying to understand the space $\text{Emb}(B,M)$ of smooth embeddings of $B$ into $M$. ...
9
votes
1answer
506 views

moduli spaces are kahler?

I often heard from experts that "moduli spaces are Kahler". This sounds as a meta-theorem asserting that every time one defines reasonable moduli spaces, then there is a standard strategy to see ...
7
votes
0answers
148 views

Why is it hard to obtain improved $L^6$ bound of eigenfunction of Laplacian on 2-dimensional compact Riemannian manifold?

Consider the $L^p$ estimate of the Laplacian on a compact boundaryless Riemannian manifold, suppose that $-\Delta_ge_{\lambda}=\lambda^2e_\lambda(x), x\in M$. C.D. Sogge proved that we have the ...
3
votes
3answers
197 views

Mechanisms generating free subgroups of Artin braid groups

Within my limited experience, I have only known free groups to occur through two mechanisms: as fundamental groups of trees (graphs) and ping-pong. And sometimes only through one way: the fact that ...
4
votes
2answers
178 views

pseudo-Anosovs with given action on homology

It is well-known that the Mapping Class Group of a closed surface of genus $g$ surjects onto $Sp(2g, \mathbb{Z})$ (see, for example the Farb-Margalit book). However, I was wondering if there is a ...
11
votes
2answers
269 views

The homeomorphism problem for hyperbolic 3-manifolds and the virtual Haken theorem

If $N$ and $N'$ are two closed hyperbolic 3-manifolds, then one would like to have an algorithm which determines whether or not $N$ and $N'$ are homeomorphic. If $N$ and $N'$ are Haken, then such an ...
9
votes
1answer
219 views

Is there a reasonable definition of TQFTs for n-cobordisms with connected inputs/outputs?

A question that's been on my mind for a while is whether any precise statement to the effect of "Heegaard Floer homology is a TQFT," for some reasonable definition of TQFT, can be made. Of course, a ...
2
votes
1answer
115 views

Are harmonic maps quasiconformal at the boundary of hyperbolic spaces?

Let $M$ be a hyperbolic $3$-manifold and $X$ a negatively curved, simply connected space with geometric boundary $\partial X$. If $\rho:\pi_1M\rightarrow Isom(X)$ does not fix a point in $\partial ...
4
votes
3answers
229 views

Domination of length functions of trees with equal covolume

(This is a reformulation of an earlier unanswered question. I would like to thank Ian Agol for pointing out to me Walter Parry's characterization of hyperbolic translation length functions.) Let $G$ ...
7
votes
2answers
329 views

Simplicial replacements in smoothing theory

As far as I can tell, ever since Milnor's Microbundles and differentiable structures (1961) paper, whenever people talk about $Diff(\mathbb R^n)$ or $PL(\mathbb R^n)$ or $Homeo(\mathbb R^n)$, they ...
9
votes
2answers
273 views

Isotopy extension theorems

I'm looking for the origins of the isotopy extension theorem in categories other than the smooth category. Precisely, in the smooth category, the isotopy extension theorem says that if $f : [0,1] ...
0
votes
1answer
85 views

Connecting two hypersurfaces in R^{n+1} by embedded curves

Let $M^n$ be a smooth closed embedded hupersurface in $\mathbb R^{n+1}$. Denote by $D$ the bounded connected component of $\mathbb R^{n+1}\backslash M$. We assume that $\mathbb R^{n+1}\backslash D$ is ...
0
votes
1answer
142 views

discrete subgroups of the isometries of a product

Suppose $X_1$ and $X_2$ are two nice metric spaces, e.g. two Riemannian manifolds, and let $G_i=Isom(X_i)$. Then $G_1\times G_2\subset Isom(X_1\times X_2)$. Suppose $X_1\times X_2$ is not compact and ...
0
votes
0answers
167 views

Fractional degree of a map?

Is there some natural notion of a fractional degree of a map? The degree of a map is a generalization of the winding number, and fractional winding numbers appear in the (mathematical physics) ...
3
votes
1answer
197 views

Symmetries of the standard probability space

The standard probability space $(I, \mathcal B, \lambda)$ consists of the interval $I = [0,1]$, its Borel $\sigma$-algebra $\mathcal B := \mathcal B(I)$ and Lebesgue measure $\lambda$. In ...
6
votes
0answers
253 views

Does a finitely generated aspherical group have an aspherical presentation with a finite generating set?

Let $G$ be a finitely generated group. Suppose $G$ has an aspherical presentation with a countably infinite generating set. Does $G$ have an aspherical presentation with a finite generating set? Here ...
1
vote
0answers
170 views

Toral decomposition

I have a couple of questions on the following theorem: Theorem. (Jaco, Shalen) Let $M$ be a compact, irreducible, orientable 3-manifold with incompressible boundary. There exists a collection ...
1
vote
0answers
137 views

Can a compact metrizable space be determined by its Hausdorff measures?

Suppose that $(X,d)$ is a compact metric space. Now suppose that $h:[0,a]\rightarrow[0,b]$ is a continuous function with $h(0)=0$ where if $x\leq y$, then $h(x)\leq h(y)$. Then define ...
6
votes
1answer
211 views

Characterisation of Q-rank 1

I'm looking for a reference and/or the original source for the following fact: An irreducible non-uniform lattice in a semisimple Lie group without compact factor has Q-rank 1 if and only if it does ...
15
votes
3answers
550 views

What is the state of the art for algorithmic knot simplification?

Question: Given a `hard' diagram of a knot, with over a hundred crossings, what is the best algorithm and software tool to simplify it? Will it also simplify virtual knot diagrams, tangle diagrams, ...
2
votes
1answer
277 views

On complex surfaces with Kodaira dimension 1

Let $S$ be a complex surface of Kodaira dimension $1$ and $\pi_{1}(S) \neq 1 $. What is known on possible diffeomorphism types of such $complex$ surfaces with a given fundamental group? Is it true ...
1
vote
1answer
230 views

Covering seifert manifolds

Let $M$ be a 3-manifold with boundary. If $M$ has an orientable finite cover that is a Seifert fiber space, then is $M$ also a Seifert fiber space?