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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0
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90 views

Help understanding a proof in Stallings' Triangle of Groups paper

I'm trying to understand the proof of theorem 1 in Stalling's Non-positively Curved Triangle of Groups. I have specific questions, but is there anywhere someone has written out the proof in more ...
0
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0answers
70 views

moduli space of equivariant holomorphic embeddings into the quintic

I'd like to understand better the following problem, whether it is mathematically well-posed, trivial, etc. Fix a non-negative integer $g$ and consider the space ...
3
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3answers
293 views

undergraduate handle decomposition. Reference

As the title says, I'm searching for a nice textbook for introducing the theory of handle decomposition of manifolds to undergraduate students.
3
votes
0answers
120 views

Lie Group Isomorphisms

I'm not sure of the difficulty of the question I'm about to ask. If it does not fit the criteria for this site then I apologize in advance, I'm rather new here. So here it goes: Let $G$ be a Lie ...
3
votes
1answer
73 views

Transfinite sequence of contiguous simplicial maps

Recall that two simplicial maps of (abstract) simplicial complexes $f,g\colon K\to L$ are contiguous if $f(\sigma)\cup g(\sigma)$ is a simplex of $L$ for every simplex $\sigma\in K$. Contiguous ...
5
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0answers
102 views

Does every null-homologous surface bound, part deux

This is a follow-up to this question, and Mark Grant's excellent answer. The answer shows that the answer is yes, but what if one were under a strange compunction to actually construct the submanifold ...
4
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1answer
112 views

0-homologous surface bounds

Given a map $f : S \to M^4$ from a compact closed not necessarily connected oriented surface to a compact oriented 4-manifold, such that $f_*([S])$ is zero in $H_2(M)$, is there a compact oriented ...
11
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1answer
170 views

Wanted: a nontrivial weakly inadmissible Heegaard diagram

This is a question asked by a student in my lecture. After drawing pictures for awhile, I thought it was a good one. I seek a nontrivial example of a pointed Heegaard diagram ...
0
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0answers
32 views

Finding the lift of a curve under some assumptions

Let $f:\Omega\subseteq\mathbb{R}^n\to\mathbb{R}^n$ be a Lipschitz function and $h$ be a vector in $\Omega$. Assume that $0\in\Omega$ and $f(0) = 0$. Also, let $\sigma:[0,1]\to\mathbb{R}^n$ be the ...
3
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1answer
168 views

Immersed Seifert surfaces of minimal genus

Let $K\subset S^3$ be a knot. We denote by $X=S^3\setminus \nu K$ the knot exterior, i.e. the complement of an open tubular neighborhood of $K$. An immersed Seifert surface for a knot $K$ is an ...
15
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2answers
408 views

Does a small-area sphere in a 3-manifold bound a small ball?

Let $M$ be a closed riemannian 3-manifold. I think that the following fact should be true and should have a relatively simple proof, but I cannot figure it out. For every $\varepsilon>0$ there ...
0
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1answer
133 views

from Dehn twists to surgery diagram [closed]

Assume the relation $(ab)^6=1$, for $a$ and $b$ Dehn twists about the meridian and the longitude of a torus. Now if we glue the two ends of $T\times I$ together by either the diffeomorphism $(ab)^6$ ...
2
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0answers
172 views

What kinds of manifolds admit non-vanishing vector fields defining convergent congruences?

One of the corollaries of the Poincaré–Hopf index theorem is that a closed, connected manifold $M$ admits non-vanishing vector fields iff its Euler characteristic is zero; i.e. $\chi(M) = 0$. I am ...
2
votes
2answers
111 views

Graph embedding in 3D grid minimizing edge length

I know that arbitrary graphs can be embedded trivially in $\mathbb{R^3}$ and that planar graphs can be drawn on a plane using Schnyder's grid embedding algorithm after triangulation. And then there is ...
8
votes
2answers
225 views

The boundary of a domain whose interior is diffeomorphic to the ball

We know that there are compact manifolds with diffeomorphic interiors but their boundaries are not homeomorphic (see the question Manifolds with homeomorphic interiors). My question is about a very ...
5
votes
1answer
204 views

Boundaries of relatively hyperbolic groups

When the interior of an n-manifold $M$ has a pinched negative curvature metric of finite volume, then its fundamental group $\Gamma=\pi_1M$ is relatively hyperbolic relative to the parabolic groups ...
10
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0answers
173 views

Fox re-imbedding theorem in dimension four

Fox re-imbedding theorem states the following: A compact 3-manifold $M$ with boundary that embeds in the three-sphere $S^3$, can be re-imbedded in $S^3$ so that its complement is a union of ...
1
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1answer
124 views

Clarification of Gabai's exposition of Murasugi Sums in 'the Murasugi sum is a natural geometric operation'

Gabai states that the Murasugi sum of two hopf bands yields a spanning surface of either the figure eight knot, the trefoil knot or a link of three components. Figure one shows two oppositely twisted ...
2
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1answer
94 views

Ray-Singer torsion of compact 3-manifolds with finite abelian fundamental group

Is the Ray-Singer analytic torsion for an arbitrary compact 3-manifold with finite Abelian fundamental group equivalent to the Ray-Singer analytic torsion of S^3 mod some direct product of Z_N's? It ...
6
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0answers
116 views

Does the shortest path between two braids pass through string links?

One of the fundamental facts underlying the application of braid theory to knot theory is that braids inject into string links. This means that braids $B_1$ and $B_2$, considered inside a cube $I^3$, ...
1
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0answers
75 views

PL or projective PL map on the links of a PL manifold

Let $M$ be a PL manifold and $f: M\rightarrow M$ be a PL homeomorphism. Suppose that $f(x)=x$ for some vertex $x$. Is the restriction map of $f$ on the links of $x$ also PL? Someone claims that this ...
8
votes
1answer
183 views

Virtual fibering conjecture for cusped hyperbolic manifolds

I am interested in understanding if the Virtual Fibering Theorem holds in the non-compact case. Agol proved that every closed hyperbolic $3$-manifold has a finite index cover which fibers over the ...
7
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1answer
449 views

Is this knot invariant already treated somewhere in the literature?

Fix a knot type $K \subset S^3$, and consider the set $$Y_K = \{ \mbox{Diagrams of }K \} / \mbox{planar isotopy}.$$ We can turn $Y_K$ into a metric space by considering the distance induced by ...
0
votes
1answer
148 views

Torelli group of a punctured elliptic curve

Let $T_{g,n}$ be the Torelli group of a $n$-punctured surface $S=\overline{S}\setminus\{x_1,\ldots,x_n\}$, with $\overline S$ orientable, closed and of genus $g$. By definition, $T_{g,n}$ is the ...
-1
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2answers
163 views

Let be $f \in Diff(M)$. What we can say about the subgroup $span{f}< Diff(M)$? What are implications in the structure of $f$ and $M$? [closed]

Let be $f \in Diff(M)$. When is finite the subgroup $span\{f\}< Diff(M)$? What are implications in the structure of $f$ and $M$?
4
votes
0answers
57 views

Convex subsets of infinite dimensional spaces up homeomorphism

Let $C$ be a convex, infinite-dimensional, non-locally-compact subset of a separable Frechet space. If $C$ is a closed subset (or more generally, if $C$ is completely metrizable), then it is known ...
2
votes
2answers
593 views

Kenji Fukaya's Lecture series at Simons center

In the past decade, theory of Kuranishi structures on moduli space of pseudo-holomorphic curves has been in the center of debates between some mathematicians in the field of symplectic geometry. ...
5
votes
1answer
191 views

Relation between Milnor ring and middle dimensional homology of hypersurface

I have suspected that the following is well-known: If $P$ is a homogeneous polynomial of degree $d$ in $n$ variables (for example, Fermat quintic $x_1^5 + \cdots + x_5^5$). The Milnor ring is ...
3
votes
2answers
157 views

Handlebody decomposition of a 3-manifold adapted to a link

Given a compact connected 3-manifold $M$ with non-empty boundary, and a link $L \subset M$, is there a handlebody decomposition of $M = H^0 \cup (\cup_i H^1_i) \cup \{\text{2-handles}\}$ such that: ...
6
votes
1answer
253 views

How many distinct homeomorphism classes of lens spaces are there with a fixed p?

This question is about the topological classification of lens spaces. Fix $p$ a positive integer, not necessarily a prime. From Brody, The topological classification of lens spaces, Annals of Math. ...
11
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3answers
537 views

Space of embeddings of circle in a surface

Let $S$ be a compact oriented surface of genus at least $2$ (possibly with boundary). Let $X$ be a connected component of the space of embeddings of $S^1$ into $S$. Question : what is the ...
3
votes
1answer
200 views

Is any smooth homeomorphism isotopic to a smooth embedding?

Let $f:D^m\to \mathbb{R}^m$ be a smooth map ($D^m$ is the unit ball). We call $f$ embedding if it is a homeomorphism on the image and the derivative $D_xf$ is nonsingular at each point $x\in D^m$ ...
4
votes
0answers
80 views

Ribbon knot presentations

Suppose $K$ is a $n$-knot, $n\geq 2$, which bounds two different ribbon disks $D_1, D_2$. These ribbon disks induce unique ribbon $(n+1)$-knots $K_1, K_2$ respectively. Is it known whether $K_1$ and ...
1
vote
1answer
135 views

Each coadjoint orbit of a compact connected Lie group $G$ admits a $G$-invariant generalized complex structure

I am looking for a proof or counterexample for following assertion Each coadjoint orbit of a compact connected Lie group $G$ admits a $G$-invariant generalized complex structure (In sense of ...
10
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1answer
192 views

Rank and hyperbolic volume

Suppose $M$ is a hyperbolic $3$-manifold whose fundamental group has rank $r.$ What is the best (lower) bound on the volume of $M?$ Similar question for rank of $H_1.$ There is a bunch of papers of ...
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
134 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
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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
272 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
106 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
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2answers
284 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
211 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
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0answers
98 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
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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
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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
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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
282 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
238 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
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1answer
157 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 ...