# All Questions

Tagged with gt.geometric-topology 3-manifolds

244 questions

**2**

votes

**0**answers

60 views

### 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 ...

**2**

votes

**0**answers

87 views

### 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 ...

**8**

votes

**3**answers

228 views

### 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 ...

**3**

votes

**2**answers

178 views

### 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 ...

**4**

votes

**1**answer

184 views

### 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)...

**1**

vote

**1**answer

118 views

### 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 $...

**9**

votes

**3**answers

575 views

### 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 ...

**5**

votes

**1**answer

177 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 ...

**12**

votes

**0**answers

182 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-...

**7**

votes

**1**answer

201 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)...

**8**

votes

**2**answers

325 views

### 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 ...

**4**

votes

**1**answer

123 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 ...

**9**

votes

**1**answer

432 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 ...

**4**

votes

**2**answers

258 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 ...

**5**

votes

**2**answers

595 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 $\pi_1(M)$, ...