Questions tagged [3-manifolds]
A three-manifold is a space that locally looks like Euclidean three-dimensional space
623
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7
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3
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Does there exist a Dehn filling of an irreducible 3-manifold with toroidal boundaries which is still irreducible?
Let $M$ be a compact, orientable, irreducible 3-manifold with incompressible toroidal boundary (there might be more than one boundary component). Is it always possible to choose appropriate slopes on ...
- 367
12
votes
2
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277
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Property P and R for general 3-manifolds
Let $Y$ be a closed oriented $3$-manifold and $K$ be a knot in $Y$. We say $K$ is the unknot if $K$ is contained in a local $3$-ball in $Y$ and is unknotted therein.
Generalized Property R:
If a Dehn ...
- 470
3
votes
1
answer
103
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Seifert invariants for Brieskorn manifolds $\Sigma(p,q,r)$
I've been studying Brieskorn manifolds $\Sigma(p,q,r)$ for $p,q,r\geq 2$. I know they are defined as the intersection of the complex surface $z_1^p+z_2^q+z_3^r=0$ as a subset of $\mathbb{C}^3$ and $S^...
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2
votes
0
answers
27
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Topological meaning of a "totally recurrent" 1d foliation in 3-manifold
I'm trying to understand Sullivan's "cycles for the dynamical study..":
https://www.math.stonybrook.edu/~dennis/publications/PDF/DS-pub-0033.pdf
which I find very complicated being ...
- 111
4
votes
0
answers
145
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Question on the construction of transversely oriented foliation on a sutured 3-manifold
The question is based on the proof of the main theorem of Gabai's paper on Foliations and the topology of 3-manifolds which is the following:
Theorem 5.1. Suppose $M$ is connected, and $(M,\gamma)$ ...
3
votes
1
answer
241
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If the complement of a knot $K$ fibers over the circle is $K$ necessarily fibered?
Let $K \subseteq S^3$ be a knot in the $3$-sphere and assume there exists a smooth map $p \colon S^3\setminus K \to S^1$ which is a fiber bundle.
For every point $\require{enclose} \enclose{...
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5
votes
2
answers
321
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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-...
- 173
1
vote
1
answer
102
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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 ...
- 841
5
votes
0
answers
89
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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 ...
- 367
17
votes
2
answers
512
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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 ...
- 331
5
votes
1
answer
374
views
Linking number and intersection number
Consider a disjoint union of two circles $A$ and $B$ smoothly embedded in $\mathbb{R}^3$ with linking number more than $1$. Suppose we know that there exists a disc $D$ in $\mathbb{R}^3$ such that $\...
- 125
7
votes
2
answers
297
views
Existence of a surface group ensures the existence of a $\pi_1$-injective immersed surface
The question is simple:
For a $3$-manifold $M$, if $\pi_1(M)$ contains a surface group $\Gamma$ (i.e. the fundamental group of some surface) then $M$ contains a $\pi_1$-injective immersed surface $S$....
4
votes
1
answer
192
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Residual finiteness and a gluing problem
The below flowchart is from Thurston's paper Hyperbolic structures on 3-manifolds I. I don't know if I interpreted it correctly but at the bottom it says that
Residual finiteness "implies" ...
2
votes
0
answers
108
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Are oriented-$h$-cobordant lens spaces orientation-preservingly homeomorphic?
Consider two three-dimensional lens spaces $N_1=L(p,q_1)$ and $N_2=L(p,q_2)$, and assume that there is an oriented-$h$-cobordism between them. In other words, we assume that there is an oriented four-...
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6
votes
1
answer
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Euler number of a Seifert bundle as a generalization of an Euler number of a circle bundle over a surface
In classic, Euler numbers associated to circle bundles over a fixed surface classify all possible such bundles. But the construction of Euler class in general requires the fact that any fiber bundle ...