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 ...

learn more… | top users | synonyms (1)

4
votes
1answer
88 views

Determining Tightness/Overtwistedness of Contact Structure using Lift of Structure to the Universal Cover

All: I would appreciate any ideas, refs., etc. on the following: Let $M^3$ be a contact 3-manifold, and let $X$ be its universal cover. Then the contact structure, say $\eta$ on $M^3$ lifts to a ...
8
votes
1answer
141 views

Mapping class group vs automorphism group in cobordism category

Let $3Cob$ be the category whose objects are closed surfaces and whose morphisms are diffeomorphism classes of cobordisms. By sending a diffeomorphism $\phi$ of a surface $X$ to its associated ...
2
votes
0answers
16 views

compact centralisers in maximal Kac-Moody groups over finite fields

Suppose that I want to choose an element of a maximal Kac-Moody group over a finite field which has a compact centraliser. Is there anything known at all about how I can go about choosing such an ...
3
votes
2answers
276 views

Automorphisms of Surfaces, Open Books and Contact Structures

Let $S$ be a surface with non-empty boundary $\partial S$ and let $f$ be an element of MCG(S), the Mapping Class Group of S, i.e., the group of self-diffeomorphisms of S up to isotopy, but with $f$ ...
2
votes
0answers
107 views

Foliation of surface all of whose leaves are circles

I'm having trouble locating a reference for the following basic fact. Let $S$ be a compact orientable surface with boundary. Assume that $\mathcal{F}$ is a foliation of $S$ all of whose leaves are ...
2
votes
2answers
286 views

Origin of the name “Torelli group”

The genus $g$ Torelli group $I_g$ is the kernel of the action of the mapping class group of a genus $g$ surface on the first homology group of the surface. The first paper I am aware of that uses the ...
10
votes
1answer
214 views

Construction of the Casson invariant

What is the easiest construction of the Casson invariant? The original construction using representation spaces (as found, for instance, in Akbulut-McCarthy) is very technical since you have to ...
1
vote
1answer
95 views

Module structure of the abelianization of the commutator subgroup

Let $G$ be a (non-abelian) group, and let $G_2$ denote its commutator subgroup. Then the abelianization $G_2^{ab} = H_1(G_2,\mathbb{Z})$ is a module over the group ring $\mathbb{Z}[G^{ab}]$. The ...
3
votes
0answers
99 views

Rigidity vs Super-rigidity of representations (of Kähler/surface groups)

In the literature there are several definitions of "rigidity" (or "super-rigidity") of representations, adapted to the circumastances. I wonder what are the relations between them; I excuse in advance ...
0
votes
0answers
110 views

When the leaves of a distribution are compact

Let $(M,\omega)$, be a symplectic manifolds. A subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if $P$ is Lagrangian P involutive ...
0
votes
0answers
65 views

Existence of a sequence of (almost) Moore irregular graphs embedded on closed surfaces

Let $S_{g}$ denote the genus $g$ closed orientable surface. I'm interested in disproving the existence of a certain configuration of simple closed curves on $S_{g}$. I'd be happy to go into more ...
1
vote
1answer
175 views

Counting edges in embeddable CW-complexes in R^3

Using Euler's formula ($V-E+F = 2$ where $V$, $E$ and $F$ are the number of vertices, edges and faces), we can easily count the number of edges in maximal graphs that are embeddable in plane: 3n-6. I ...
1
vote
1answer
296 views

How many types of jigsaw puzzle pieces in n dimensions?

I was partitioning jigsaw puzzle pieces with some friends yesterday and we noticed that there are 6 types of pieces: All 4 sides have a knobby bit sticking out 1 side has a knobby bit sticking out 2 ...
3
votes
2answers
243 views

Realizing homology classes on surfaces with boundary by simple closed curves

Let $\Sigma$ be a compact oriented surface with boundary. Assume that the genus of $\Sigma$ is positive. We say that an element $h \in H_1(\Sigma)$ can be realized by a simple closed curve if there ...
0
votes
1answer
158 views

An example for the case tht if the leaves of polarization be non-compact then the polarized sections are not square integrable

Let $(M,\omega)$, be a symplectic manifolds. A subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if $P$ is Lagrangian P involutive ...
2
votes
1answer
192 views

Going from _Actual_ Open Books to _Abstract_ Open Books

All: I am looking for a proof of the result that any open book $(B, \pi)$ ; $B$ a fibered link , $\pi$ a map of $M^3-B \rightarrow S^1$ on a 3-manifold $M^3$ , so that $\pi$: $(M^3-B)$ fibers over ...
6
votes
2answers
158 views

Are there some list of the finite subgroups of the mapping class groups of low genus surfaces?

We already know the bound of the order of the finite subgroups of the $Mod(S_g)$. If we take a further step, to find all the finite subgroups, then what is the result for low genus cases? For example, ...
2
votes
0answers
46 views

What groups of symmetry are most suited for filling uniformely a spherical 3D space, whilst possessing the lowest possible surface-to-volume ratio?

I am looking for the closest known approximate solution to Kelvin foams problem that would obey a spherical symmetry. One alternative way of formulating it: I am looking for an equivalent of ...
1
vote
0answers
89 views

Entangled helical knots

Consider a pair of disjoint, congruent helices $H_1$ and $H_2$ passing through one another in the following sense. (Caveat lector: This question is not of general interest! It is also long.) $H_1$ is ...
4
votes
0answers
91 views

are smooth homotopic open embedding of Hilbert manifolds smoothly isotopic?

A theorem of Chapman (Chapman, T. A. Homotopic homeomorphisms of infinite-dimensional manifolds. Compositio Math. 27 (1973), 135–140) states that every two open embeddings $f,g:M\to N$ of Hilbert ...
8
votes
3answers
318 views

What constant ensures hyperbolicity of Dehn surgery?

I am interested in showing that certain knots having a surgery description are hyperbolic. Unfortunately I have not had time yet to read Thurston's work, so my understanding of this is vague. But from ...
2
votes
1answer
140 views

Frohman & Fine's proof about Bianchi groups as HNN extensions (or anyone else's)

Specifically, I am looking for a proof that for squarefree $d\in\mathbb{N}\setminus\{ 1,3\}$, there exists some Kleinian $\Gamma$ and some $\alpha\in$Aut$(PSL(2,\mathbb{Z}))$, such that the Bianchi ...
7
votes
0answers
174 views

Reference Request: Topological h-cobordism theorem in higher dimensions

I think this question on math.stackexchange is more appropriate on mathoverflow. Correct me, if you don't think so. The h-cobordism theorem is true in the topological and in the smooth category in ...
2
votes
2answers
204 views

Action of Mapping Class Group on Arc complex

Suppose $S$ is a surface of finite type with nonempty boundary. Now consider the arc complex $\mathcal{A}$. The action of Mod(S)(mapping class group) on the set of all vertices has finitely many ...
-1
votes
1answer
141 views

$S^n$ admit a real polarization $D\subset TS^n$?

When the $n$-sphere, $S^n$,admit a real polarization $D\subset TS^n$
6
votes
2answers
490 views

All mapping space between CW complexes is a CW complex?

Let $\mathrm{Map}(X,Y)$ denote the (unbased) cellular mapping space from $X$ to $Y$. If $X$ and $Y$ are finite CW complexes, is $\mathrm{Map}(X,Y)$ a CW complex? Can we know the cell structure of ...
0
votes
1answer
203 views

A problem related to connectivity of analytic functions

Let $f(z)\in A(\mathbb D)$, where $A(\mathbb D)$ is the space of analytic functions on the open unit disk $\mathbb D$ and continuous on $\overline{\mathbb D}$. Question: Is the connectivity of ...
-1
votes
1answer
133 views

Homeomorphism of the punctured sphere which fixes an essential Jordan curve

$\phi$ is a homeomorphism from the 2-sphere to itself which represents an element of $PMCG(S^2,A)$ (we also denote it by $\phi$), where $A$ is a finite set of $S^2$. $\gamma$ is an essential Jordan ...
7
votes
1answer
182 views

High dimensional generalized Poincare hypothesis without the h-cobordism theorem?

The generalized PL Poincare hypothesis states that in dimension $n$ there is a unique PL manifold that has the homotopy-type of $S^n$. It's known to be true in all dimensions except perhaps $n=4$. ...
5
votes
1answer
116 views

Equivariant smoothing of PL structures on $S^3$

Suppose $S^3$ is PL sphere on which a finite group $G$ acts by PL homeomorphisms. Is it always possible to find a compatible smooth structure such that $G$ acts by diffeomorphisms? I am not quite ...
4
votes
1answer
171 views

Codimension zero embeddings and diffeomorphism groups

Let $V$ be a smooth manifold obtained by attaching the ``open collar'' $[0,1)\times \partial N$ to a compact smooth manifold $N$ along the boundary. Let $\mathrm{Emb}(N, V)$ be the space of embeddings ...
2
votes
0answers
51 views

Minimal set of 2-2 Pachner move null sequences on a (nonplanar) trivalent graph?

A "null sequence" is of course a sequence of Pachner moves (inside a closed area) that doesn't change the trivalent graph. E.g. doing the same Pachner move twice (leads to orthogonality of 6j symbols) ...
4
votes
2answers
278 views

Who first used the cross-ratio to describe shapes in hyperbolic geometry?

I was reading this Wikipedia article today:https://en.wikipedia.org/wiki/Shape#Similarity_classes and I realized that it strongly resembles the use of coss-ratios as "shape parameters" in hyperbolic ...
8
votes
1answer
202 views

Hyperbolic 3-manifolds fibering over the circle

Suppose you have the mapping torus $M_\phi$ of some pseudo-Anosov map $\phi.$ Is there some sufficient or necessary condition on $\phi$ to assure that $M_\phi$ has large injectivity radius? I am aware ...
9
votes
6answers
1k views

Is there a good notion of morphism between orbifolds?

Following Thurston, an orbifold is a topological space which looks locally like a finite quotient of $\mathbb R^n$ by a finite group of $O(n)$: this is expressed using charts as for differentiable ...
10
votes
1answer
518 views

The mathematics of tavern puzzles

I remember seeing a paper on the arxiv this year (which I cannot now find Edit: This paper: http://arxiv.org/abs/1208.6545, found by j.c.) proposing to study the linkage of rigid bodies such as tavern ...
4
votes
1answer
185 views

The cyclic branched covers of “simple” knots in $S^3$

Is there a convenient place in the literature where the geometric decompositions of cyclic branched covers of $S^3$ branched over "small" knots is recorded? By small knots, I'm referring to things ...
2
votes
1answer
132 views

Can a tree cover a finite graph with arbitrary large girth?

Let $G$ be some discrete finitely generated group acting cocompactly on a leafless tree $T$. Is it true that for any natural number $n$ there is a finite graph $\Gamma$ such that: $T$ is the universal ...
20
votes
1answer
771 views

A function composed with itself produces the identity

Let $B$ be the closed unit ball in $\mathbb R^3$ and $f: B\to B$ continuous, such that $f\circ f$ is the identity (i.e., $f\circ f=\mathbb 1_B$) and $f$ restricted on $\partial B$ is also the identity ...
-2
votes
2answers
229 views

topology on varieties

Let $X, Y$ be varieties over $\mathbb{C}$, and the topology I am talking about is the Eucliden topology. I am not sure if the following two results are true, and where can I find the references: (1) ...
5
votes
0answers
123 views

List of cubical spaces

Suppose I have a three-dimensional cube (I tend to think of it as a regular ideal cube in $\mathbb{H}^3,$ but you don't have to). I glue up its sides in some way to obtain topological spaces. The ...
5
votes
2answers
287 views

Thurston-Cannon $S^2$-filling curves

I have been looking into equivariant space-filling curves $S^1\to S^2$ as discussed by Cannon & Thurston in these two papers: Three-Dimensional Manifolds, Kleinian Groups and Hyperbolic Geometry ...
6
votes
1answer
138 views

Why is the dividing set nonempty when a convex surface has Legendrian boundary?

I am an undergrad and curious about the following question. Let $(Y,\xi)$ be a contact manifold, and $L\subset (Y,\xi)$ be a Legendrian knot which is the boundary of a convex surface $\Sigma$ embedded ...
4
votes
1answer
243 views

What is known about maximal free subgroups of surface groups?

Let $\Gamma_g=< a_1,...,a_g,b_1,...,b_g | \prod_{i=1}^g [a_i,b_i]>$ (a surface group). What is known about maximal free subgroups of $\Gamma_g$ for $g>1.$ (I.e. free subgroups which are not ...
10
votes
1answer
296 views

Tverberg's theorem in CAT(0) spaces

Does Tverberg's theorem hold for CAT(0) spaces of covering dimension $d<\infty$: Is it true that for any $d$-dimensional $CAT(0)$-space $X$ and a subset $E\subset X$ of cardinality $(d + 1)(r - ...
23
votes
8answers
2k views

Are there some other notions of “curvature” which measure how space curves?

I am learning differential geometry and have a few questions on curvature. -- Background: Gauss invented "Gauss curvature" to measure how surface curves. Riemann gives an ingenious generalization ...
2
votes
2answers
109 views

Is there a relative Pachner theorem?

Consider an $n$-dimensional PL manifold $P$ together with a closed subpolyhedron $P_0$ such that there exists a finite combinatorial triangulation of $P$ that restricts to a triangulation of $P_0$. ...
2
votes
0answers
77 views

Characterizations of product groups under quasi-isometry

This is a follow-up question to Quasi-isometric rigidity of certain products of groups. It can be moved to a comment if it is too trivial. If all the asymptotic cones of a finitely generated group ...
4
votes
1answer
101 views

Quasi-isometric rigidity of certain products of groups

Let $G_1,G_2$ be two delta-hyperbolic groups. If $G_1\times \Bbb{Z}$ is quasi-isometric to $G_2\times \Bbb{Z}$, must it be true that $G_1$ is quasi-isometric to $G_2$? This is similar to the classic ...
7
votes
1answer
164 views

Is there a combinatorial version of PL ambient isotopy in dimension $>3$?

The classical PL Reidemeister Theorem reads: Reidemeister Theorem: Two knots in $S^3$ are PL ambient isotopic if and only if any diagram of one can be transformed into a diagram of the other by ...