The tag has no wiki summary.

learn more… | top users | synonyms

4
votes
1answer
196 views

Mapping torus relative to an infinite orbit can be hyperbolic with finite volume?

Consider a homeomorphism of the sphere with an infinite orbit converging forwards and backwards to the same point. Remove the orbit and the accumulation point and make the mapping torus. Can the ...
18
votes
1answer
903 views

Diffeomorphisms vs homeomorphisms of 3-manifolds

For a smooth 3-manifold $M$, is the natural map from the space of diffeomorphisms of $M$ to the space of homeomorphisms of $M$, $${\sf Diff}(M) \longrightarrow {\sf Top}(M),$$ a weak homotopy ...
5
votes
1answer
660 views

Dehn surgery on handlebody

Assume $V$ is a handlebody and $C$ be a simple closed curve contained in the interior of $V$. As Sam said, there exists some simple closed curve such that every dehn surgery along it produces a ...
4
votes
1answer
277 views

what is the meaning of “inseparable” in this case

Let $i:T^{2}\rightarrow S^{1}\times S^{2}$ be an embedding map. If $i(T^{2})$ is inseparable in $S^{1}\times S^{2}$, then $S^{1}\times S^{2}-i(T^{2})\cong (O\cup K)^{c}$. Here $(O\cup K)^{c}$ is a two ...
5
votes
3answers
508 views

Questions on 3-manifolds with a given boundary

I have the following question: For a given two-dimensional Riemann surface $C$, Is there a way to classify all topologically distinct three-dimensional compact manifolds $M$ whose boundary is $C$, ...
1
vote
1answer
332 views

Smooth structures on closed $3$-manifolds are unique up to diffeomorphism?

Hi! I'm using the theorem stated in the question, but so far, I haven't found a source that does explicitely state and prove it or at least give a proper citation. It must be me though, since the ...
2
votes
3answers
585 views

what is the meaning of a curve $C$ representing Identity in fundamental group?

Suppose $M$ is a closed 3-manifold, $C\subset M$ is a simple closed curve which represent identity in $\pi_{1}(M)$. Then $C$ bounds an immersed disk in $M$. My question is: When does it bound an ...
12
votes
2answers
654 views

How to see isometries of figure 8 knot complement

The figure 8 knot complement $M$ is the orientable double cover of the Gieseking manifold, which implies that $M$ has a fixed-point free involution. If we think of $M$ with its hyperbolic metric, this ...
0
votes
1answer
124 views

is very compact P^2-irreducible 3-manifold homotopy equivalent to a sphere or cell-quotient?

is very compact $P^2$-irreducible 3-manifold homotopy equivalent to a sphere or cell-quotient?
2
votes
1answer
271 views

The Tubular Neighborhood of a Closed Geodesic

Suppose $M_{g}$ is the mapping torus $\Sigma_{g} \times [0, 1]/ (x, 0) \equiv (\tau x, 1)$, where $\Sigma_{g}$ is the hyperbolic space with genus $g,$ and $\tau : \Sigma_{g} \to \Sigma_{g}$ is an ...
6
votes
2answers
291 views

$S^1$-action in three dimensions

Let $M$ be a closed, orientable 3-manifold with a non-trivial differentiable $S^1$-action. What does this imply for $M$? What are examples except for (products of) spheres?
8
votes
1answer
460 views

Topological rigidity of compact manifolds in dimension three

The Borel Conjecture asserts that homotopy equivalent aspherical closed manifolds are homeomorphic, which is still open in general. But, for three-dimensional manifolds, this conjecture holds (I ...
1
vote
1answer
103 views

Will the rank of fundemantal 3 manifold be decreased is I module the n(n>1) times of a element?

I am doing some rank control about the fundamental group of a 3 dim hyperbolic orbifold. After cuting out the regular neighborhood of all the singularity, I get a manifold with imcompressible ...
8
votes
1answer
1k views

Incompressible surfaces in an open subset of R^3

Let $U$ be a connected open subset of $\mathbb R^3$. Furthermore, we have: $\mathbb R^3\setminus U$ has exactly two connected components (thus by Alexander duality, $H_2(U;\mathbb Z)=\mathbb Z$). ...
4
votes
1answer
362 views

What is known rigoruously about the semiclassical (k to infinity) limit of WRT invariants?

To each closed $3$-manifold $N$, there is a corresponding Witten--Reshetikhin--Turaev invariant $Z_k(N)$ depending on an integer $k$ (the level) and a Lie group $G$ (and perhaps we'll just concentrate ...
3
votes
3answers
588 views

Polynomial Vector Fields on the 3-Sphere

EDIT(3): I am looking for a basis for the Lie algebra of polynomial vector fields on $S^3$. EDIT(2): I am fairly certain now that my question is more along the lines of, what does the Lie algebra of ...
13
votes
1answer
633 views

Where are $+$, $-$ and $\infty$ in bordered Heegaard-Floer theory?

Here goes my first MO-question. I've just read Lipshitz, Ozsváth and Thurston's recently updated "A tour of bordered Floer theory". To set the stage let me give two quotes from this paper. ...
4
votes
2answers
336 views

Does a homeomorphism of $S^1 \times S^2$ which is homotopy to the identity has to isotope to it?

I guess the question can be asked for all manifolds. But I am particularly interested in $S^1 \times S^2$ right now. Concrete example preferrd.
10
votes
2answers
418 views

Embedding the product of three circles in the 4-sphere.

Let $M$ be a smooth submanifold of the 4-sphere $S^4$. I'm going to demand that $M$ be diffeomorphic to $S^1 \times S^1 \times S^1$. By Jordan-Brouwer separation, $M$ separates the 4-sphere into ...
27
votes
0answers
1k views

What is the three-dimensional hyperbolic volume of a four-manifold?

Every smooth closed orientable 4-manifold may be constructed via a handle decomposition. Before asking a couple of questions, I recall some well-known facts about handle-decompositions of 4-manifolds. ...
7
votes
1answer
298 views

3-orbifolds with a Seifert geometry that are not actually Seifert fibered

It is well-known that Seifert fibered $3$--manifolds are geometric: they admit one of the Thurston geometries $S^2 \times R$, $R^3$, $H^2 \times R$, $S^3$, $Nil$, and $PSL(2,R)$. Furthermore, the ...
3
votes
3answers
614 views

Is a compact, connected, orientable 3-manifold with $\mathbb{Z}^K$ fundemental group uniquely determined?

According to the Kneser-Milnor prime decomposition theorem for 3-manifolds, any compact, connected, orientable 3-manifold $M$ is diffeomorphic to $S^3 / \Gamma_1$ # $\cdots$ # $S^3/ \Gamma_n$ # $(S^2 ...
5
votes
1answer
577 views

Trace identities and the Kauffman Bracket skein module

Let's consider $K_t(M)$, the Kauffman bracket skein module (see this and this papers) of a three-manifold $M$. When $t=-1$, $K_t(M)$ is easily seen to be isomorphic to the ring of functions on the ...
3
votes
1answer
305 views

A question about Dehn filling in the unknot.

Hello, If you have a link of 2 components $U$ and $V$ in $S^3$. $U$ is the unknot, and you make $1/q$ dehn filling in $U$, you can visualize the resulting knot $V'$ from $V$, by twisting it $q$ ...
3
votes
0answers
308 views

What is the behaviour of a smooth 3-manifold acting by a circle?

As Mumford pointed out in his paper 'Topology of Normal Singularities and a Criterion for Simplicity'(1961), every point $p$ on a normal complex surface $V$ has an associated 3-manifold $M$ which is ...
5
votes
2answers
707 views

Residual Finiteness of Fundamental Group of Compact 3-Manifold

Hempel, in his 1987 article "Residual Finiteness for 3-Manifolds", shows that if $M$ is a compact Haken 3-manifold, then $\pi_1(M)$ is residually finite. The outline of the proof is basically: ...
5
votes
1answer
413 views

Generalization of Moise's theorem

I am looking for a generalization of Moise's theorem, which the few professors that I asked treat as a "known geometric fact" but none could find a reference to an article proving it. The claim is ...
7
votes
1answer
569 views

Geometry of Whitehead manifolds.

I'm currently studying some problems about the Whitehead manifold $W$ (the open 3-manifold which is contractible but not homeomorphic to $\mathbb{R}^3$). Does there exists some survey paper on its ...
1
vote
2answers
407 views

Do homeomorphisms of boundary components of 3-manifolds extend to the manifold?

The question that I'd like to answer can be generalized to the following: if $M$ is an orientable 3-manifold and $F$ is a boundary component of $M$ (which may have other boundary components), can an ...
3
votes
2answers
565 views

Fundamental groups of closed hyperbolic 3-manifolds are freely indecomposable

I believe the following statement is true, and I've even seen it referenced here. Could someone point me to a proof? The fundamental group of a closed hyperbolic 3-manifold is not a free product. ...
4
votes
2answers
699 views

Heegaard splitting, equivalent homeomorphisms, mapping class group of genus n-torus

Given a Heegaard splitting of genus $n$, and two distinct orientation preserving homeomorphisms, elements of the mapping class group of the genus $n$ torus, is there a method which shows whether or ...
8
votes
2answers
430 views

Do the results of (1/n)-surgery determine the link?…

Knowing the result of knot surgery is often not enough to determine the knot. Indeed, there are 3-manifolds admitting an infinite number of descriptions as surgery on a (1-component) knot in $S^3$. ...
14
votes
3answers
592 views

Maximal euler characteristic of surfaces bounding two fixed curves

Let $\gamma_0$ and $\gamma_1$ be two simple closed curves in a closed surface $S$. What is the maximum Euler characteristic of a compact properly embedded surface $\Sigma \subset S\times [0,1]$ ...
9
votes
1answer
308 views

Ramified cover of 3-ball

I it true that any connected oriented 3-manifold with nonempty boundary can be obtained as a ramified cover of 3-ball with a ramification at a link? link = a 1-dimensional submanifold with possibly ...
13
votes
1answer
1k views

Detecting a cover of the figure-8 knot complement

I have a specific concrete example of a complete, finite volume, cusped hyperbolic 3-manifold $M$, and I am trying to determine whether or not $M$ is a cover of the figure-8 knot complement (call this ...
6
votes
5answers
391 views

In a contact manifold, is every tranverse 1-foliation given by some Reeb vector field?

Let $M$ be a contact manifold, and let $F$ be an oriented 1-dimensional foliation that is transverse to the contact structure. Is there a contact form $\alpha$ whose associated Reeb vector field ...
9
votes
1answer
545 views

Lower bound on number of tetrahedra needed to triangulate a knot complement

Following along a similar line to the question asked here: Is there an explicit bound on the number of tetrahedra needed to triangulate a hyperbolic 3-manifold of volume V? Let $K$ be a (hyperbolic) ...
10
votes
3answers
690 views

Is the cut locus of a generic point in a hyperbolic manifold a generic polyhedron?

Let $p\in M$ be a point in a closed riemannian manifold $M$. Recall that the cut locus of $p$ is the subset of $M$ consisting of all points that are connected to $p$ by at least 2 distance-minimizing ...
0
votes
1answer
347 views

pseudo-Anosov maps on surfaces with boundary

In "Automorphisms of Surfaces after Nielsen & Thurston" by Casson & Bleiler (on pages 75 - 80) they discuss classifying automorphisms of a surface. They show that, if $S$ is a closed ...
22
votes
4answers
2k views

Can all n-manifolds be obtained by gluing finitely many blocks?

Fix a dimension $n\geqslant 2$. Let $S= \{M_1,\ldots, M_k\}$ be a finite set of smooth compact $n$-manifold with boundary. Let us say that a smooth closed $n$-manifold is generated by $S$ if it may ...
8
votes
1answer
442 views

Is the spectrum of closed geodesics in a closed hyperbolic 3-manifold asymptotically homogeneously dense?

It seems to me that the results in this important paper of Kahn-Markovich imply the following fact. Let $M^3$ be any closed hyperbolic 3-manifold. For every $\epsilon > 0$ there is a natural number ...
5
votes
2answers
351 views

Unknotting tunnels in surface bundles

Given a once-punctured surface $F$ and an orientation preserving self homeomorphism $\phi$, let $M_\phi$ be the bundle over $S^1$ with fiber $F$ and monodromy $\phi$. In Sakuma's survey article The ...
1
vote
1answer
352 views

Why do strongly irreducible Heegaard surfaces look like fibers?

I remember hearing somewhere that strongly irreducible Heegaard surfaces in hyperbolic 3-manifolds "look like" fibers. I know that Otal's result about short geodesics in hyperbolic mapping tori being ...
6
votes
2answers
1k views

Totally geodesic surfaces in fibered 3-manifolds

Is there an easy example of a (closed) hyperbolic 3-manifold that fibers over the circle but contains some totally geodesic surface? (Of course such manifolds exist if the 'Virtually Fibered ...
10
votes
2answers
830 views

Geometrization for 3-manifolds that contain two-sided projective planes

Often when people write about the geometrization conjecture they assume (for simplicity) that the manifold is orientable. I never seriously thought of non-orientable 3-manifolds, but while reading ...
24
votes
2answers
2k views

Why is the volume conjecture important?

The volume conjecture, a formula relating hyperbolic volume of a knot complement with the semiclassical limit of a family of coloured Jones polynomials, is widely considered the biggest open problem ...
8
votes
2answers
781 views

Simplifying triangulations of 3-manifolds

Throughout, by finite triangulation I mean a triangulation consisting of a finite number of triangles. Suppose $T$ and $T'$ are finite triangulations of a 3-manifold $M$. We will say that $T'$ is ...
5
votes
2answers
605 views

Intutive interpretation about Linking forms

Let $M^3$ be a rational homology 3-sphere. (i,e, $M^3$ is closed 3-manifold with $H_{*}(M;Q)=H_{*}(S^3;Q)$ As beautifully explained in Ranicki's Algebraic and Geometry surgery book and Davis-Kirk's ...
17
votes
2answers
1k views

Is the 4x5 chessboard complex a link complement?

The 2x3 and 3x4 chessboard complexes (form a square grid of vertices and make a simplex for any set of vertices no two of which are in the same row or column) are a 6-cycle and a triangulated torus ...
21
votes
3answers
1k views

How expensive is knowledge? Knots, Links, 3 and 4-manifold algorithms. [closed]

With geometrization, Rubinstein's 3-sphere recognition algorithm and the Manning algorithm, 3-manifold theory has reached a certain maturity where many questions are "readily" answerable about ...