# Questions tagged [3-manifolds]

A 3-manifold is a space that locally looks like Euclidean 3-dimensional space

412
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### Alexander duality for Homology sphere which is the Geometric realization of a finite simplicial complex

The Alexander duality Theorem is usually stated for a triangulable pair $(\mathbb S^n, Y)$ where $Y$ is a subset of the standard sphere $\mathbb S^n$. My question is: Does the duality also hold if we ...

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### Smoothening pseudo-Anosov flows

A topological Anosov flow on a closed 3-manifold can be replaced by a smooth Anosov flow using an argument of Fried: use Markov partitions to find a surface of section, put in other terms, one can ...

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### Maximally symmetric hyperbolic 3-manifolds with finite volume

In physics, standard cosmology is build with simple maximally symmetric 3-manifolds (spacelike time-slices of constant curvature, e.g. $S^3$ or less popular the hyperbolic space $H^3$). Since $S^3$ ...

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### Totally geodesic submanifolds of SO(3) [closed]

Consider the special orthogonal group $SO(3)$ with its bi-invariant metric (or equivalently, with the metric induced by its standard embedding to the space of $3\times 3$ real matrices).
Obviously, $...

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### Bounds on convergence of two orbits in the limit set of a Schottky group

Suppose we have two points in the limit set of a Schottky group, $x,y\in \Lambda(\Gamma)$. Consider the orbits of those points under a primitive subset $\Gamma'$ of $\Gamma,$ that is, one that does ...

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### Integer surgery on $S^3$

I know that any compact orientable 3-manifold can be obtained from the three sphere $S^3$ by an integer surgery. I am not sure why the surgery operation is completely determined by Where we map ...

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### Two Dehn fillings yielding a lens space?

Let $M$ be an oriented $3$-manifold with $\partial M$ torus. Suppose that two different Dehn fillings $M(r)$ and $M(r')$ are (oriented) homeomorphic to a lens space $L(p,q).$ Does that imply that $M$ ...

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### Hyperbolic manifolds with infinite cyclic fundamental group

It is a well known fact that there is a correspondence between complete hyperbolic $n$-manifolds up to isometry and discrete subgroups of isometries of the hyperbolic space $\mathbb{H}^n$ that act ...

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### Understanding fundamental group of Poincare homology sphere

I'm currently reading Knots, Links, Braids, and 3-Manifolds by V. V. Prasolov and A. B. Sossinsky. I have trouble understanding the following picture. The dashed line denotes a trefoil whose tubular ...

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### Does every $SL_2\mathbb{C}$ representation of a closed oriented surface extend over a compact oriented three-manifold?

Let $F$ be a compact oriented surface and $\rho:\pi_1(F)\rightarrow SL_2\mathbb{C}$ be a representation. Does there exist a compact oriented three-manifold $M$ with $\partial M=F$ and a homomorphism $...

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### Approximative extension of the autohomeomorphism of the complement of the trivial knot?

Let $S^1\subset \mathbb{R}^3$ be the unit circle and suppose $h\colon \mathbb{R}^3\setminus S^1\to \mathbb{R}^3\setminus S^1$ is a homeomorphism. Clearly it might be that $h$ cannot be extended to $S^...

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### Contractibility and orientation double cover

Question. Let $M$ be a triangulated non-orientable 3-manifold with non-orientable boundary. (It is possible to assume that the boundary is the Klein bottle.) Let $\ell$ be a non-orientable loop on the ...

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### Cobordism and Kirby calculus

It may be a simple question but I wonder to ask:
Is it possible to draw a homology cobordism between $3$-manifolds by using the techniques of Kirby calculus?
At least, for instance, Brieskorn ...

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### Kirby calculus on Mazur manifolds

I have questions about Akbulut and Kirby's paper Mazur manifolds.
I couldn't figure out the following equality passages:
Any help will be appreciated.

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### The plumbing graphs of Brieskorn spheres

Let $p,q$ and $r$ be positive integers. A Brieskorn sphere is a closed oriented $3$-manifold defined by $$\Sigma(p,q,r) = \{ x^p+y^q+z^r=0 \} \cap S^5.$$
Its fundamental group is well-known due to ...

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### Heegaard splittings of Brieskorn spheres

The genus $g$ handlebodies are building blocks of $3$-manifolds. They are constructed from $3$-ball $B^3$ by adding $g$-copies of $1$-handles $B^2 \times B^1$. Their boundaries are homeomorphic to the ...

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### Generalized Schoenflies - formalizing step in proof?

[Sorry if the level here is wrong, I asked this on math.SE, but even with a bounty, it got no attention.]
I am currently reading Hatcher's 3-Manifolds notes, the part proving Alexander's theorem, ...

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### Long non-deformable hyperbolic fillings

The title and question have been edited in light of Ian Agol's comment. The previous question was stated in terms of the wrong notion of length to discuss deformations:
What is the longest slope $\...

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### Implications of Geometrization conjecture for fundamental group

Hempel proved that Haken manifolds have residually finite fundamental groups. With the Geometrization conjecture, this now holds for any compact and orientable 3-manifold.
How exactly does the ...

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### Residual Finiteness for 3-Manifolds Hempel

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. In the proof he starts by reducing the case to ...

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### Aspherical manifold with superperfect fundamental group and non-trivial center?

I am interested in knowing if there is a closed, (smooth) aspherical manifold $M$ (hyperbolic would be best) with superperfect fundamental group (that is to say, with $H_1(\pi_1(M);\mathbb{Z}) = H_2(\...

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### Reconciling Sullivan's theorem with the hyperbolic structure of the Figure–8 knot complement

I am interested in the 3-manifolds with hyperbolic structures from the physics (gravity) perspective. I encounter this paper https://arxiv.org/pdf/hep-th/9812206.pdf whose Eq. (9) mentions a theorem ...

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### Searching for a Thurston paper with egg / 3-manifold analogy?

I remember coming across some article of Bill Thurston’s where he describes a 3-manifold (with boundary?) as being like an egg. In my recollection the interior of the egg, the shell, and even the ...

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### Genus 2 3-manifolds bounding only $X^4$ with $b_2(X^4)$ big?

The genus of a closed orientable 3-manifold $M^3$ is the minimum genus among all Heegaard splitting surfaces for $M$. Every such 3-manifold bounds a compact 4-manifold. Let $I(M)$ denote the ...

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### Group of parallelizations of $M^3$ finitely generated?

Let $M^3$ be a compact orientable 3-manifold. Then $TM$ is trivial and let's go ahead and fix a trivialization $\tau : M \times \mathbb{R}^3 \to TM$. Then given a map $g : (M, \partial M) \to (SO(3),...

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### Does Heegaard splitting relate topological properties of a $3$-manifold to properties of subgroups of $MCG$

In the proof of Lickorish-Wallace theorem, we use Heegaard splitting of a closed, orientable and connected $3$-manifold and obtain a surface diffeomorphism which glues the two handle-body. I wonder ...

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### Two papers on surface diffeomorphisms

The following two papers appeared in the reference of a paper i was reading.It seems that neither is published formally.Is there a website where i could find them?
A. Casson, Cobordism Invariants of ...

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### What's the “actual” shape of a black hole accretion disk?

[Warning: I have no expertise in general relativity, so this question might not be very rigorous]
More and more often we come across science popularization articles like this one which show beautiful ...

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### Simply put Floer homology

I would like to understand what exactly Floer homology $HF_*(Y)$ for a simply connected compact 3-manifold $Y$ is. I understand there are many variants of Floer homology (and cohomology) but I would ...

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### Minimum number of relations that must be added to make a group abelian

Let $G$ be a finitely generated group and let $c(G)$ denote the minimal number of relations that we must add to a presentation fro $G$ in order to make $G$ abelian. I would like to find examples of ...

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### Reference request: A knot is tame if and only if it has a tubular neighbourhood

Definitions:
A knot is an embedding $\kappa:S^1\hookrightarrow S^3$ (we do not require smooth or polygonal).
Two knots $\kappa,\,\lambda:S^1\hookrightarrow S^3$ are equivalent if one of the following ...

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### A coincidence between the Lambert cube, Lobell polyhedron, and hyperbolic 3-manifolds?

I. Lambert cube $\mathfrak L(\alpha_1,\alpha_2,\alpha_3)$
In this paper (p.8), we find the volume $V$ of the hyperbolic Lambert cube for the special case $\alpha=\alpha_1 = \alpha_2 = \alpha_3$ as
$...

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### Does WRT invariant detect hyperelliptic involution on the genus 2 surface?

The Witten-Reshetikhin-Turaev invariant cannot detect the hyperelliptic involution on the genus 1 surface, and that if $M_U$ is the mapping torus for a mapping class group element $U\in \mathrm{Mod}(\...

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### Obtaining the bounding 4-manifold from the Heegaard diagram

It is well known that any orientable closed 3-manifold $M$ admits an Heegaard splitting $M = H_1\cup H_2$ where $H_i$ is an handlebody of genus $g$. It is also well known that such an $M$ is the ...

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### Prerequisites for Hempel's 3 manifolds?

What are the prerequisites for Hempel's 3 manifolds book? I've heard you need a good amount of PL topology. I have read Jennifer Schulten's book and found it accessible, but I've heard Hempel is tough....

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### Are Turaev-Viro invariants holonomic?

Consider a 3-manifold $M$ with a boundary, which is a genus $g\geq 1$ surface $\Sigma$. Fix a triangulation $T$ of $\Sigma$. Then Turaev-Viro invariants $TV_q(M)$ are functions, assigning to integer ...

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### Quotient of Three Dimensional Torus by Permutation on Coordinates

The Mobius Strip can be realized as a quotient of $T = (S^1)^2$ via the identifications $(x,y) \sim (y,x)$.
I tried to generalized this concept to a higher dimension, and consider the quotient of $(...

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### Are triangulations with common refinements PL-homeomorphic?

Do there exist simplicial triangulations $K_1$ and $K_2$ of a topological manifold $M$ such that $K_1$ and $K_2$ have a common subdivision but they are not PL-homeomorphic? Ideally, I would like an ...

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### What are these 3-manifolds from surgery?

I know that surgery on the unlink with +0 slope gives $S^2 \times S^1$ (where all the links above are embedded in $S^3$). I figured (I think) that surgery on the hopf link (with +0 on both) returns $S^...

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### Circle bundles and surface bundles which admit no strongly irreducible Heegaard splittings

Let $S$ be a closed connected orientable surface with $g(S)>0$. Jennifer Schultens, in her paper ``The Classification of Heegaard Splittings for (Compact Orientable Surface)$\times S^1$'', proves ...

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### SnapPy isometry routine

Dear Colleagues and Friends,
Here's a question that I hope some of you, more experienced in programming, can answer.
Once SnapPy is used to compute the symmetry group of a hyperbolic manifold by ...

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### Existence of smooth structures on topological $3$-manifolds with boundary

It is said in this thread Unique smooth structure on $3$-manifolds that every topological $3$-manifold admits a smooth structure. However it is not specified whether the manifolds are allowed to have ...

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### open book decompositions of $\Sigma\times S^1$

Let $\Sigma$ be a closed orientable surface. Is there a standard open book decomposition on the $3$-manifold $M=\Sigma\times S^1$?
If the binding is allowed to be empty in the definition of an open ...

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### Triangulations of 3-manifolds in Regina and SnapPy

I have been doing some statistical studies on small 3-manifolds, and I note that one can produce larg-ish censuses of triangulations in Regina. Now, the Regina documentation tells us how to convert a ...

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### P-adic Volume Conjecture

Let $M$ be a closed hyperbolic 3-manifold. One can use hyperbolic structure on $M$ to define hyperbolic volume $Vol(M)$. Thanks to Mostow's rigidity theorem the volume depends only on the topology of ...

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### Hyperbolic Dehn surgeries and SU(2)-representations

Let $S^3-K$ be the complement of the figure eight knot complement. Thurston, in his Lecture Notes, constructed a hyperbolic structure, which comes from a discrete, faithful representation $\pi_1(S^3-K)...

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### Signature/nullity function for a link obtained by parallel pushoffs of a knot?

Let $K$ be an oriented knot in $S^3$ together with a framing $n$. Let $K(a,b)$ be the oriented link obtained by taking $a$ copies of the $n$-pushoff of $K$ with the same same orientation as $K$ and $...

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### 3d Chern-Simons TQFT of gauge group (E8)$_1$ = SO(16)$_1 \otimes$ a trivial spin TQFT = Cartan E$_8$ matrix

In this post, we like to relate the following 3 bosonic TQFTs that can be defined on generic non-spin manifold $M^3$.
Given a non-abelian Chern-Simons (CS) TQFT of a gauge group $G$ and the $k$ named ...

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### Manifold with no closed components?

Let $M$ be a manifold with boundary. Reading some papers on $3$-manifolds I have come across some statements where they require that: ”$M$ has no closed components.”
What does this mean? The ...

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### What is variation of the Chern-Simons functional, and why can it be calculated as follows?

Let $G$ be a Lie group. Assume that we have an Ad-invariant bilinear symmetric form $$\langle-,-\rangle : \mathfrak{g} \times \mathfrak{g} \to \mathbb{C}.$$ Given a smooth manifold $X,$ we let $\...