Questions tagged [3-manifolds]
A three-manifold is a space that locally looks like Euclidean three-dimensional space
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A graph manifold without an orientation reversing involution?
Is there a graph manifold (https://en.wikipedia.org/wiki/Graph_manifold) that doesn't admit an orientation reversing involution? If so, what would be a simple example?
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Stein fillable tight contact structures on the 3-torus
Kanda classified tight contact structures on the 3-torus. Which of them is Stein fillable? Is there any good reference?
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Extension of smooth structure on three dimensional topological manifolds
Let $M$ be a three dimensional compact topological manifold and $U$ an open set in $M$ homeomorphic to some smooth $C^k$ manifold $U'$, $1 \leq k \leq \omega$. Can we extend the smooth structure on $U$...
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Formula for the Casson invariant in terms of the linking form
The paper 'Trisections, intersection forms and the Torelli group' by Peter Lambert-Cole quotes the following formula for the Casson invariant of a knot $K$ in a homology $3$-sphere in terms of the ...
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Reference request: Stallings-Epstein-Waldhausen construction
I am looking for a reference for the Stallings-Epstein-Waldhausen construction (constructing an incompressible surface in a 3-manifold from a nontrivial splitting of the fundamental group).
I know of ...
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Ideal triangulation of hyperbolic 3-manifold with generic mapping class group
I am from physics background so I apologize in advance if my question is trivial.
Kojima proves for every finite group $G$, there is a hyperbolic 3-manifold such that its mapping class group equals $G$...
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Expositions of Stallings's fibration theorem
In his famous paper
Stallings, John,
On fibering certain 3-manifolds. 1962 Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961) pp. 95–100 Prentice-Hall, Englewood ...
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Two details from Stallings's proof of the sphere theorem
EDIT: After a little prompting by Mark Grant, I answered the first question in the comments. The second question remains open.
Let $M$ be a compact $3$-manifold with $\pi_2(M) \neq 0$. The sphere ...
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Is the number of prime factors of 3-manifolds obtained by Dehn surgery along a link with $N$ components in $S^3$ bounded from above?
For a given $N$, is the number of prime factors of 3-manifolds obtained by Dehn surgery along a link with $N$ components in $S^3$ bounded from above?
The Two Summands Conjecture states that surgery ...
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Euler characteristic of pseudomanifolds with boundary
It is a well-known fact that for every compact oriented odd-dimensional manifold $\mathcal{M}$ with boundary it holds that
$$\chi(\mathcal{M})=\frac{1}{2}\chi(\partial\mathcal{M}).$$
In particular, if ...
- 1,147
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votes
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answer
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Normal form of framed links under Kirby moves
It's well known that any oriented closed 3-manifold (topological or smooth) can be obtained by surgerizing along a (framed oriented) link $L$ in the 3-sphere $S^{3}$. Even better, Kirby found a ...
- 5,038
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Quadratic cusp shape
Which hyperbolic $3$-manifolds are known to have quadratic cusp shape?
Explanations: Cusps of hyperbolic $3$-manifolds are products torus x interval. They lift to horoballs in hyperbolic $3$-space, ...
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Revisiting Gordon-Luecke theorem
$\DeclareMathOperator\Diff{Diff}\DeclareMathOperator\GL{GL}$Here is an proof-sketch of a strengthened Gordon-Luecke theorem. This is presumably known, but is it written down somewhere? I am also ...
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Isotopy of open book supporting same contact structure
In dimension 3, the Giroux correspondence gives us a bijection between contact structures (up to isotopy) and open book decompositions (up to positive stabilisation). Moreover, Giroux shows that two ...
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Is there a simple formula to compute the Casson invariant of an homology $3$-sphere from its Heegaard diagram?
Let $(S_g,\boldsymbol{\alpha}_1,\boldsymbol{\alpha}_2)$ be a Heegaard diagram of a Heegaard splitting $\Sigma=H_g \cup_{\phi_1\phi_2^{-1}}H_g$ of an integral homology sphere $\Sigma$, i.e. $S_g$ is a ...
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$\pi_1(M^3)$ containing a normal infinite cyclic subgroup
Let $M^3$ be a compact $3$-manifold such that $\pi_1(M)$ contains a normal subgroup isomorphic to $\mathbb Z$.
Can we show either $\pi_1(M)$ is torsion-free or $\pi_1(M)=\mathbb Z \oplus \mathbb Z_2$ ...
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$P^2$-irreducibility of a $3$-manifold
A $3$-manifold $M$ is called $P^2$-irreducible if it is irreducible and there is no $2$-sided $P^2$ contained in $M$.
Can we show $M$ is $P^2$-irreducible iff $\pi_2(M)=0$?
Notice that one direction ...
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Do once-punctured torus bundles have integral traces?
Once-punctured torus bundles are a well-studied class of hyperbolic 3-manifolds, but unfortunately I have been unable to find out whether they always have integral traces (in the sense of [1], Def. 5....
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Ambiguity in the unoriented knot connected sum
It is well known that there might be ambiguity in the unoriented knot connected sum if the knots concerned are not invertible.
E.g., consider 8_17, the only knot with crossing number 8 which is non-...
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3-manifold with boundary containing a projective plane
Let $M$ be a compact $3$-manifold such that no component of $\partial M$ is $S^2$ and one component $F$ of $\partial M$ is the projective plane.
If $i_*:\pi_1(F) \to \pi_1(M)$ is an isomorphism, can ...
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Special kind of 3-manifolds
Is there an open connected orientable 3-manifold $M$ with the following properties:
$M$ admits a complete hyperbolic metric with finite hyperbolic volume.
$H_{i}(M,\mathbb{Z})=0$ for any $i>0$.
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One-sided incompressible surface in 3-manifolds
Let $M^3$ be a closed orientable $3$-manifold. If $H_2(M,\mathbb Z)=0$ and $H_2(M, \mathbb Z_2)\ne 0$, can we show that $M$ contains a 1-sided incompressible surface?
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The moduli space of finite volume hyperbolic 3-manifolds?
By finite volume hyperbolic 3-manifold, I do mean $M=\mathbb{H}^{3}/\Gamma$ where $\Gamma$ is a torsion-free Kleinian group such that the hyperbolic volume $Vol(M)<\infty$.
I will call
$$\mathcal{M}...
- 153
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Noncompact three-manifold with fundamental group isomorphic to a surface group
Let $M$ be an open orientable three-manifold such that $\pi_1 (M)$ is isomorphic to the fundamental group of a closed orientable surface $S\ncong \mathbb{S}^2$. Furthermore, suppose that $\tilde{M} \...
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Classifying nested 3-manifolds with fundamental group property
Let $M_1\subseteq M_2\subseteq\mathbb R^3$ be closed connected subsets with smooth boundary. Suppose that every closed loop in $M_1$ is freely homotopic inside $M_2$ to a closed loop inside $M_2\...
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Surgery diagrams of 3-dim abstract open books
Starting with an abstract open book $(\Sigma,\phi)$, I would like to understand some of the manifolds that I could obtain. Given a surface $\Sigma$ and monodromy $\phi$, it is not hard to find a Kirby ...
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Are there non-homeomorphic 3-manifolds with the same Turaev-Viro-Barrett-Westbury invariants?
The Turaev-Viro-Barrett-Westbury invariant of a closed oriented topological $3$-manifold $M$ for a spherical fusion category $\mathcal{C}$ is a number denoted $|M|_{\mathcal{C}}$ computed from (but ...
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Balanced presentation of the fundamental group of a Seifert fiber space
Is there any readily available reference for a balanced presentation of the fundamental group in terms of the classifying invariants of an arbitrary Seifert fibered space? I can't find it, and the ...
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Transverse open book decompositions supporting the same contact structure
An open book decomposition on an oriented 3-manifold $M$ is a fibered oriented link $B\subset M$, bounding a foliation by Seifert surfaces $\Sigma_t \subset M$, $t\in S^1$, $\partial \Sigma_t = B$.
A ...
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3-manifold proof of Grushko's theorem
Grushko's theorem says that given an epimorphism $\phi: F \to G_1 * G_2$ where $F$ is a finitely-generated free group, there exists. subgroups $F_1$ and $F_2$ of $F$ so that $F = F_1 * F_2$ and $\phi(...
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Circle-valued Morse function and minimal genus
I think the following two statements are true, and most likely are in the literature. If so, could someone point me to some references? If not, counterexamples?
Let $Y$ be a closed oriented connected ...
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Thurston universe gates in knots: which invariant is it?
Today I discovered this nice video of a lecture by Thurston:
https://youtu.be/daplYX6Oshc
in which he explains how a knot can be turned into a "fabric for universes". For example, the unknot ...
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What's the Milnor's link group for the trivial knot in a lens space?
For a link $L$ in a 3-manifold $Y$, Milnor's paper "Link Groups" https://link.springer.com/content/pdf/10.1007/BF01393902.pdf defined the link group as some quotient of $\pi_1(Y-L)$. If $L$ ...
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Smooth Schoenflies theorem for compact $3$-manifolds
Let $M^3$ be a compact $3$-manifold with $\partial M=N$ a connected surface. Suppose one has a smooth embedding of $N$ into the interior of $M$ and $N$ bounds a domain $D$ in $M$. Can we show that $D$ ...
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Surface in a product domain
Let $M$ be a connected closed surface. Suppose $N$ is a connected closed surface embedded in the interior of $M \times [0,1]$ such that $N$ separates $M \times \{0\}$ and $M \times \{1\}$. Can we ...
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3-dimensional h-cobordisms
Let $W$ be a $3$-dimensional $h$-cobordism of closed surfaces $M_0$ and $M_1$. Can we prove that $W$ is trivial? That is, $W$ is homeomorphic to $M_0 \times [0,1]$.
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2
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Tightness/Overtwistedness of genus one open book decomposition
Suppose we have an open book decomposition $(P,\phi)$ of a 3-manifold $Y$, where $P$ is a punctured torus and $\phi$ is the monodromy. We know $\phi$ can be represented by a matrix in $SL(2,\mathbb{Z})...
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Open cone homeomorphic to the Euclidean space
Let $X$ be a topological space and the open cone $C(X)$ over $X$ is defined to be $X \times [0,1)$ with $X \times \{0\}$ identified. Suppose $C(X)$ is homeomorphic to $\mathbb R^4$, can we prove that $...
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Reference sought for Seifert fiber spaces
I seek a reference for what is surely a well known basic result about Seifert fibered 3-manifolds. Namely they are all obtained by Dehn-surgery along a regular Seifert fiber (and the surgery slope is ...
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Do different decompositions of Dehn twists induce the same cobordisms between mapping tori?
Suppose $Y=T^2\times I/f$ is a mapping torus, where $f$ is a diffeomorphism of $T^2$. Suppose $c$ is a curve on $T^2\times \{1\}$ with surface framing and suppose $D_c$ is the positive Dehn twist ...
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Möbius cross energy in $S^3$?
Let $\gamma_i$, $i=1,2$ be two loops in $\mathbb R^3$. The Möbius cross energy of the pair is defined by
$$
E(\gamma_1, \gamma_2)=\iint_{S^1\times S^1}\frac{|\gamma'_1(u)|\cdot|\gamma'_2(v)|}{|\...
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Presentations of mapping class groups in dimension $3$
For any closed oriented surface $M$, its mapping class group $MCG(M)$ can be generated by Dehn twists along certain curves on $M$. A presentation for the group $MCG(M)$ was found in [1] and then ...
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Mapping class group of a twisted I-bundle over $RP^2$
$\DeclareMathOperator\Mod{Mod}\DeclareMathOperator\Homeo{Homeo}$Let $\Mod(M)=\pi_0(\Homeo(M))$ be the mapping class group of a manifold, possibly with boundary (I'm including the orientation reversing ...
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Functoriality of Thurston's norm
Let $M$ be a manifold of dimension $3$ and let $N$ be an embedded submanifold of $M$ (also of dimension $3$).
Then, both second homologies $H_2(M)$ and $H_2(N,\delta N)$ are equipped with a norm (...
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Zagier's "From 3-manifold invariants to number theory"?
Zagier lectures on "From 3-manifold invariants to number theory" - do you know about texts of that or on the discussed web of ideas? ([https://www.mpim-bonn.mpg.de/de/node/10791])
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Is finding boundary-reducing discs for PL 3-manifolds with boundary pattern computationally efficient?
I am working with manifolds with boundary pattern, as defined by Matveev in his book Algorithmic Topology and Classification of 3-Manifolds. A manifold with boundary pattern is a pair $(M, P)$ where $...
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The Kirby diagram of a manifold glued along the lens space $L(p,1)$
Suppose $K$ is a knot in $S^3$ with any framing and $m=m_0$ is its meridian with $-1$ framing. Suppose $m_1,\dots,m_{p-1}$ are unknots with framings $-2$, such that $m_{i-1}$ and $m_i$ are linked as a ...
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Boundary map in Mayer-Vietoris sequence of cohomology
Suppose $M$ is a 3-manifold with connected boundary. Let $T$ be a tangle in $M$, i.e., $T$ is a embedded connected 1-submanifold whose boundary is on $\partial M$ ($T$ is not closed). Moreover, ...
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Decidability of knot equivalence in general 3-manifolds? Surface equivalence?
Given a closed orientable 3-manifold $M^3$ and two knots $K_1$ and $K_2$ in $M$, is there an algorithm to decide if $K_1$ and $K_2$ are isotopic? Is there an algorithm to decide if there is a ...
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References on Hyperbolic Geometry and Teichmuller Theory
I am asking a soft question here.
I am learning hyperbolic geometry on my own. Recently, I have completed the book "Fuchsian Groups" by Svetlana Katok. Also, I have background in Lee's three ...
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