# Questions tagged [3-manifolds]

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

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### Examples of incomplete Lorentz 3-manifolds

Reading this paper where closed 3-dim. Lorentz manifolds with noncompact isometry groups are studied, I wonder if all of them are geodesically complete.
One class of 3-dim. closed Lorentz manifolds ...

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**1**answer

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### Pre-images of Seifert surfaces are incompressible?

Consider a knot $K \subset S^3$ and let $M_K$ be the associated double branched cover. The pre-image $S$ of a Seifert surface is a surface without boundary inside $M_K$.
Can $S$ be incompressible? If ...

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### Reference request for wild 3-manifolds

I’m looking for a text on 3-manifolds that focuses on wild/pathological objects, similar to Bing’s work in the field. I know basic algebraic topology (homotopy, homology, cohomology) and have read ...

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148 views

### Is there a generalized Property P - what can we say about framed link descriptions of $S^3$?

A knot $K$ is said to have Property P if every nontrivial Dehn surgery on $K$ yields a 3-manifold that is not simply connected. It is known that every knot except the unknot has Property P. I am ...

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131 views

### Are there exotic twisted doubles of 4-manifolds?

Take a smooth 4-manifold $X$ whose boundary has a diffeomorphism $\tau: \partial X \to \partial X$ that extends to a homeomorphism but not a diffeomorphism of $X$. (By Matveyev and Curtis-Freedman-...

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191 views

### Action of diffeomorphism group on non-vanishing vector fields

Let $M$ denote a closed manifold. Let $\Gamma(TM\setminus 0) $ denote the space of non-vanishing sections of $TM$. Note that the diffeomorphism group $\text{Diff} (M)$ acts on $\Gamma(TM\setminus 0)...

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### Hyperbolic $3$-manifold groups that embed in compact Lie groups

Is there a closed hyperbolic $3$-manifold whose fundamental group is isomorphic to a subgroup of some compact Lie group?
It is known that every surface group can be embedded into any semisimple ...

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208 views

### Lorentzian analogue to Thurston geometries

Is there an analogue to the eight Thurston geometries for Lorentz metrics?
If so, how many "disctinct" geometries are there in the Lorentzian case?
And which closed 3-manifolds admit metrics which ...

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118 views

### Simple invariants to detect concordance in general 3-manifolds

Let $Y$ be a closed, connected, orientable 3-manifold. We call to oriented knots $K_1, K_2$ in $Y$ (smoothly) concordant if there is a smoothly, properly embedded annulus in $Y \times I$ such that ...

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196 views

### Embedding problem for 3-manifolds attacked via 4-manifolds

In this archiv paper which is continuation of following:
Borodzik, Maciej; Némethi, András; Ranicki, Andrew, Morse theory for manifolds with boundary, Algebr. Geom. Topol. 16, No. 2, 971-1023 (2016). ...

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185 views

### Which 3-manifolds are known to admit exotic pairs of bounding 4-manifolds?

Let $M$ be a compact connected three manifold. By an exotic pair of bounding 4-manifolds, I mean two smooth 4-manifolds $X_1,X_2$ such that $X_1$ and $X_2$ are homeomorphic but not diffeomorphic, and ...

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227 views

### Open book decompositions of $T^3$

Please pardon my ignorance on the subject of open books, I'm a noob. I would like to know some explicit descriptions of open book decompositions of the three torus $T^3$. Are there examples with ...

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514 views

### Classification of closed 3-manifolds with finite first homology group?

I am interested in a topological classification of connected closed 3-manifold $M$ that have finite homology group $H_1(M)$.
Since $H_1(M)$ is the abelization of the fundamental group $\pi_1(M)$, ...

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113 views

### Possible orders of automorphisms for the Poincare homology sphere

Let $M^3$ denote the Poincare homology sphere. I am wondering what the possible orders of (smooth) automorphisms of $M$ are (I'm not sure if allowing arbitrary homeomorphisms changes things?). By ...

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242 views

### JSJ decomposition and classification of 3-manifolds

I need some philosophical explanation for JSJ decomposition theorem. It says that closed orientable irreducible 3-manifold can be cut along set of incompressible tori onto pieces which are:
atoroidal ...

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### Seifert fiberings of zero euler number which are semi-bundles

Let M be a closed oriented manifold which has the structure of a "semi-bundle" (See Section 1.2. of Hatcher's notes on three-manifolds) over an interval I. Assume that M is Seifert fibered over a base ...

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### 3-manifold being boundary of neighborhood of 2-complex in 4-space

In this question I have asked about boundary of regular neighborhood of $\mathbb RP^2$ in $\mathbb R^4$. I am interested in more general way of producing 3-manifolds in $\mathbb R^4$ namely the ...

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165 views

### Cusps in hyperbolic manifolds and fundamental group

I am reading the book "The Arithmetic of Hyperbolic Three Manifolds" by Maclachlan and Reid and I am having some problems in understanding something about cusps.
The definition they give of a cusp is ...

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680 views

### Proof of Giroux's correspondence

It is extensively used and cited the following statement due to Giroux:
Given a closed $3$-manifold $M$, there is a $1:1$ correspondence between oriented contact structures on $M$ up to isotopy and ...

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462 views

### Find a surface or 3-manifold whose fundamental group is $(\mathbb{Z}/n\mathbb{Z}) \rtimes (\mathbb{Z}/2\mathbb{Z})$

I know by Van Kampen's Theorem that we can obtain $\pi_1(S_1 \vee S_1) = \mathbb{Z} * \mathbb{Z}$, so I am wondering if we can construct a surface or 3-manifold whose fundamental group is $\mathbb{Z}...

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67 views

### Blow-up and Blow-down kirby local moves for non-orientable $3$-manifold

Can anyone explain or give a reference about the Blow-up and Blow-down Kirby local moves for non-orientable $3$-manifolds?
Thanks, advance.

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235 views

### Is the Lisca-Matic bound (aka slice-Bennequin bound) strictly stronger than the Bennequin bound?

The Bennequin bound [1] says that, for a transverse knot (or later link) $K$ in $S^3$,
$$\mathrm{sl}(K) \le - \chi(\Sigma)$$
for any Seifert surface $\Sigma$ for $K$, where $\mathrm{sl}$ is the self-...

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**1**answer

406 views

### Random 3-manifolds in $R^4$

Consider following program:
Generate random 3-manifold embedded in $R^4$.
Perform its triangulation.
Put it to Regina and calculate what manifold it is.
Assuming that we have good algorithm for ...

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**1**answer

391 views

### Non-orientable 3-manifolds

I am reading Non-orientable 3-manifolds of small complexity (Topology and its Applications 133 (2003) pp 157-178, arXiv:math/0211092), by Amendola and Martinelli. In this work $\mathbb P^2$-...

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348 views

### Lowest Dimension for Counterexample in Topological Manifold Factorization

Bing gave a classical example of spaces $X, Y, Z$ such that $X \times Y = Z$, where $X$ and $Z$ are manifolds but $Y$ isn't. The space $Z$ in his example has dimension four. Is it known if this is ...

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168 views

### Step by step construction of 3-manifolds in $R^4$

My question has survived. Therefore I try another one. Consider some elementary operations on closed compact 3-manifold $M \subset R^4$. These elementary operations are e.g. $0$-surgery or $1$-surgery ...

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**1**answer

168 views

### Are fundamental groups of web complements residually finite?

While thinking of whether any web (spatial trivalent graph) without an embedded bridge can be realized as a branching locus of a finite branched cover over $S^3$, I realized that this problem is ...

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153 views

### Intersection of $\pi_1$-injective surfaces

Let $M$ be a closed non-Haken hyperbolic $3$-manifold. Are there two, nonhomotopic, $\pi_1$-injective closed surfaces in $M$, and which do not intersect?

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### Uniqueness of spheres in prime decomposition of a 3-manifold

Let $M$ be a closed connected orientable 3-manifold. Then Kneser tells us that there is a decomposition $M = P_1 \sharp \cdots \sharp P_k$ of $M$ into prime manifolds. Milnor tells us that if $M = ...

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161 views

### Immersed incompressible surfaces in surface bundles

Let $M$ be a closed, oriented, hyperbolic $3$-manifold which is a surface bundle over $\mathbb{S}^1$.
Is there some $\pi_1$-injective closed surface (perhaps not embedded) $S \subset M$ which is not ...

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**1**answer

237 views

### How many homotopy types of lens spaces L(p,q) if the given integer p is not prime?

There is a theorem of Whitehead that lens spaces $L(p,q)$ and $L(p,q')$ are of the same homotopy type iff $\pm qq'≡ m^2 (\mathrm{mod}\ p)$ for some $m$. As a consequence, for a given $p$, there is ...

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95 views

### Can we compute the Tristram–Levine signatures of a knot in $S^3$ using Jacobian with Fox partial derivatives?

My question is in the tittle:
Can we compute the Tristram–Levine signatures of a knot in $S^3$ using Jacobian with Fox partial derivatives?
If the answer is yes, is there a reference for this.

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**1**answer

416 views

### What is “topology in dimension 3.5”?

I've noticed a couple of conference titles which reference something called
"topology in dimension 3.5," such as this one and this one. This subject seems quite mysterious to me — it looks like ...

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291 views

### General position for map from surface to 3-manifold

Let f be a smooth map from a (compact,oriented) surface S to a (compact, oriented) 3-manifold M. Suppose that I have an embedded (non-contractible) loop $\gamma$ in my surface $S$, can I find an (...

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### Nice proof of the Reidemeister-Singer’s theorem?

Is there a nice proof (preferably with pictures) of the Reidemeister-Singer theorem? I'd prefer some classical methods, perhaps in a book or lecture notes?
I want to learn how things are done.

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**1**answer

238 views

### Dehn twist generators for mapping class group of a genus zero surface with boundary

Can you help me find a reference or explain how to find explicit Dehn twist generators for $MCG(S_{0,n})$, the mapping class group of a genus $0$ surface with $n$ boundary components, fixing the ...

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99 views

### Enumeration of three dimensional spherical good orbifolds covered by Nil, sol and E3

Is there in the literature a list of three dimensional spherical, good orbifolds covered by nil, Sol and E3, and their algebraic topological invariants? (Homology, orbifold fundamental group).

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### 3-manifolds with all geodesics closed

A theorem of Bott states that if a manifold admits a metric with all geodesics closed, then its homology is isomorphic to the homology of one of the manifolds from the list: $S^n, \mathbb{RP}^n, \...

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### Can we cut and rotate a particular region of a hyperbolic 3-manifold to get another (non-homeomorphic) hyperbolic 3-manifold?

I'm trying to learn more about hyperbolic 3-manifolds, in particular the geometric implications of doing hyperbolic Dehn surgery to suitable knot complements.
Following this paper by Christian ...

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313 views

### a question about mapping class groups

According to Thurston's construction, which can be found for instance in Farb-Margalit's A Primer on Mapping Class Groups, theorem 14.1 (here is a link to the version I am using: http://www.maths.ed....

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153 views

### Can a knot be cables of two different knots?

I wonder if there is an example of a knot $K$ in the 3-sphere which can be realized as cables of two distinct (up to isotopy) knots $K_1 \neq K_2$.
It is known that if a knot $K$ is the $(p,q)$-cable ...

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**1**answer

199 views

### Different Heegaard splittings of a 3-manifold

I want to study same 3-manifolds with different Heegaard splitings.
Of course one has stabilization, but even with the same genus, we have different Heegaard splittings.
If we encode a 3-manifolds by ...

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131 views

### Lutz twist and open book decompositions

Let $M^3$ be a closed oriented 3-manifold, endowed with an open book decomposition. Consider a section of the open book, that is a knot $K \subset M$ disjoint from the binding and meeting every page ...

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**1**answer

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### Pull-back of knots in branched covers and the Alexander polynomial

Given a knot $K \subset S^3$ one can form its double branched cover $\Sigma_2(K)$ and consider the pull-back knot $\widetilde{K} \subset \Sigma_2(K)$ of $K$ to $\Sigma_2(K)$ (the locus fixed by the ...

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270 views

### Given a link $L\subset S^3$ how to construct a link $L'$ whose complement have hyperbolic structure?

Thurston claimed that almost all closed 3-manifolds are hyperbolic. To support this, he said that every closed 3-manifold is obtained by Dehn surgery along some link whose complement is hyperbolic. ...

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650 views

### Can a finite group action by homeomorphisms of a three-manifold be approximated by a smooth action?

Let $M^3$ be a smooth three-manifold, and let $\gamma:G\to\operatorname{Homeo}(M)$ be a finite group action on $M$ by homeomorphisms.
Can $\gamma$ can be $C^0$-approximated by smooth group actions $...

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### Examples of interesting non orientable closed 3-manifolds

In dimension 2, there are two remarkable non-orientable closed manifolds, the projective plane (from synthetic geometry; has the fixed point property; algebraic compactification of the plane etc) and ...

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### Question on Neumann-Zagier Symplectic matrix of an ideal triangulation of one cusped 3-manifold

For a knot complement $M\backslash K$ on a general closed 3-manifold $M$, the gluing equations are given by ($k$ is the number of ideal tetrahedra in an ideal triangulation of $M\backslash K$)
$c_I:=\...

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### About the boundary of a fibered cone of a 3-manifold

Let $S$ be a closed surface, $\psi$ a pseudo-Anosov map, $M$ be the mapping torus, $\tilde{S}$ be a $\mathbb{Z}$-fold cover of $S$ using an invariant cohomology class. Let $D$ be a fundamental domain, ...

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906 views

### Simple question on Kirby move

From hyperbolic volume computation, I found that the following two 3-manifolds are (possibly orientation-reversal) homeomorphic:
surgery on figure-eight knot $4_1$, with slope $-5$, and
surgery on $...