# Questions tagged [3-manifolds]

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

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### Non-orientable real algebraic three-dimensional manifolds

Smooth real algebraic hypersurfaces of even degree in $\mathbb{RP}^4$ that are maximal (i.e. that are homologically as rich as possible in the sense of the Smith-Thom inequality) are all non-...
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### The diameter of the projection of a convex core

Let $M$ be a closed hyperbolic 3-manifold and $H_{g}$ a genus g handlebody. Assume that $\pi: int(H_{g})\rightarrow M$ is a cover. Denote $N\subset H_{g}$ the convex core. My question is: If the ...
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### $M^3$ admits $Sol$ geometry if and only if $\pi_1M$ is virtually solvable but not virtually nilpotent?

Let $M$ be a closed, orientable, irreducible 3-manifold and having an infinite fundamental group. Is it true that $M$ admits $Sol$ geometry if and only if $\pi_1M$ is virtually solvable but not ...
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### $3$-manifold that is a surgery on a knot

By the Lickorish-Wallace theorem, every oriented closed $3$-manifold can be obtained by a surgery on a link in $S^3$. In the statement of this result, links are required: not every such manifold can ...
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### If the universal cover has three boundary components, does it have infinitely many?

Suppose that $M$ is a compact, connected three-manifold with boundary. Suppose that $\pi_1(M)$ is infinite. Suppose that $\tilde{M}$, the universal cover of $M$, has at least three boundary components....
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### Every closed surface divides a closed 3-manifold $X$ into two parts if and only if $H_1(X)$ is finite

My friend is looking for proof of the following statement Every closed surface divides a closed 3-manifold $X$ into two parts if and only if $H_1(X; \mathbb{Z})$ is finite. Rumor source: Justin ...
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### Explicit parameterizations of complicated unlinks?

I have a somewhat empirical question which I hope is still welcome here. I would like to know how to write down explicit parameterizations of "complicated unlinks", say with 2 or 10 ...
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### Covering of a knot complement

Let $B=S^3\setminus K$ for some (tame) knot $K$. Suppose we have a covering $E\to B$ with a finite fiber. Question: is $E$ homeomorphic to a knot/link complement? On this question I found only the ...
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The unknot and the Hopf link are (as far as I know) the only links whose complements have abelian fundamental groups. Are there more examples whose complement have amenable fundamental group?
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### Discreteness of volumes of boundary-parabolic representations

Suppose $M$ is a cusped hyperbolic $3$-manifold of finite volume. Let $\mathfrak{R}_0(M)$ be the space of boundary-parabolic representations $\rho : \pi_1(M) \to \operatorname{PSL}_2(\mathbb C)$. Is ...
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### Definition of Thurston's skinning map

A key construction in Thurston's proof of the existence of hyperbolic structures on Haken manifolds is the so-called "skinning map" associated to a 3-manifold $M$ with boundary whose ...
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### Example of a non $\pi_1$-injective, degree one, self-map of a three-manifold

All manifolds will be assumed to be closed, oriented, and connected. Let $f\colon M\to M$ be a map of degree $\pm 1$. It is not hard to show that $\pi_1(f)$ is surjective. What is an example of a non ...
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### Lens space bounding a topological, simply-connected 4-manifold with $b_2=1$

The following is written in section 1.6 (p.7) of this paper: https://arxiv.org/pdf/1010.6257.pdf. ($\cdots$) Which lens spaces bound a smooth, simply-connected 4-manifold $W$ with $b_2(W)=1$? ($\cdots$...
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### Non-compact three-manifolds with the same proper homotopy type are homeomorphic?

I am looking for some literature with some (counter) examples of the following fact (though I don't know if the fact is true or not): Let $M, M'$ be two non-compact connected $3$-manifolds with the ...
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### Homology of spherical $3$-manifold group

I have been studying $3$-manifolds recently and I got stuck in the following situation. For lens spaces the below fact is true. Let $G$ be a finite group acting freely and orthogonally on $S^3$ so ...
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### Ideal triangulations of $3$-manifolds with "cusps" of genus $\ge 2$

Typically when one thinks about ideal triangulations of a $3$-manifold the link of each ideal vertex is a circle, so the ideal points correspond to toroidal cusps; alternatively, one can truncate the ...
Stallings' celebrated Fibration Theorem states that if a closed irreducible $3$-manifold $M$ admits a short exact sequence $$1 \to N \to \pi_1(M) \to \mathbb{Z} \to 1,$$ ...
The bigon criterion for surfaces says that if two simple closed curves $\alpha$ and $\beta$ embedded on a surface $\Sigma$ intersect in points $\{p_1,\dotsc,p_n\}$ and $\alpha$ and $\beta$ can be ...