# Questions tagged [kirby-calculus]

Kirby diagrams of 4-manifolds, Kirby moves and Kirby calculus, Akbulut diagrams, handle decompositions

Kirby diagrams of 4-manifolds, Kirby moves and Kirby calculus, Akbulut diagrams, handle decompositions

25
questions

2
votes

1
answer

113
views

A knot in $S^3$ is called a smoothly slice knot if it bounds a smoothly embedded 2-disk in $D^4$. Every ribbon knot is known to be a smoothly slice knot, and there are known some nontrivial smoothly ...

9
votes

2
answers

399
views

Part 1: The following picture is from Saveliev's book Lectures on Topology of 3-manifolds, page 130:
He indicates that the knot drawn in the solid torus $S^1 \times D^2$ is homologous to $S^1 \times \...

5
votes

1
answer

462
views

Consider the Kirby diagram $ D$ given by a 2-component unlink, both dotted circles.
In general, when performing a 1-handle slide over another 1-handle, the band chosen must not link any dotted circle,...

5
votes

1
answer

345
views

TLDR I want to see more examples of exotic $4$-manifold (hopefully connected, simply connected, oriented, and closed).
Are there known presentations of $4$-manifolds $M$ with exotic structures, ...

1
vote

0
answers

140
views

Given a smooth manifold $M$ of dimension $n$ and a diffeomorphism $\phi: M \to M$, we can construct a smooth cobordism of dimension $(n+1)$ from $M$ to $M$ by gluing $M \times [0,1]$ with itself by $\...

4
votes

0
answers

243
views

In dimension 3, we have the celebrated Kirby theorem: Let $L_1, L_2$ be two links in the 3-sphere $S^3$; then they surgeries along them produce homeomorphic 3-manifolds if and only if they are related ...

9
votes

1
answer

691
views

Given an (oriented framed) knot $K$ in the 3-sphere $S^3$, we can perform a surgery along $K$ to get another 3-manifold $M$. From $M$, we can perform the inverse surgery back to $S^3$.
However, the ...

4
votes

1
answer

68
views

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 ...

2
votes

0
answers

141
views

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 ...

6
votes

1
answer

421
views

Akbulut's cork is the Mazur manifold $W$ shown in the picture below,
This manifold carries an involution of it's boundary $f:\partial W\to \partial W$ that exchanges a meridian of the 0-framed curve ...

12
votes

2
answers

1k
views

It may be a simple question but I wonder to ask:
Is it possible to draw a homology cobordism between $3$-manifolds by using the techniques of Kirby calculus?
At least, for instance, Brieskorn ...

10
votes

1
answer

344
views

I have questions about Akbulut and Kirby's paper Mazur manifolds.
I couldn't figure out the following equality passages:
Any help will be appreciated.

1
vote

0
answers

322
views

Let $M$ be a closed smooth submanifold of $N$. It is well known that tubular neigbourhoods of $M$ are diffeomorphic to the normal bundle of $M$ in $N$ and therefore to each other. Are they smoothly ...

3
votes

1
answer

248
views

It is well known that any orientable closed 3-manifold $M$ admits an Heegaard splitting $M = H_1\cup H_2$ where $H_i$ is an handlebody of genus $g$. It is also well known that such an $M$ is the ...

6
votes

1
answer

289
views

I'm interested in doing $2$-surgeries to $\sharp^k S^1 \times S^3$. That is to the manifold obtained from applying $1$-surgeries to $S^4$.
Since $\pi_1(O(3)) = \mathbb{Z}_2$, there are two possible ...

4
votes

0
answers

116
views

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

3
votes

0
answers

97
views

Simplicial sets, CW complexes
Simplicial sets can be described completely algebraically, by specifying a family of sets, and maps between them satisfying certain relations. This description can be ...

4
votes

0
answers

225
views

Since handle decompositions and Morse functions are intimately related, I'm imagining that a given explicit handle decomposition allows for an explicit description of the cellular complex and thus of (...

4
votes

2
answers

603
views

Let $M\colon \partial_- M \to \partial_+ M$ be an oriented, compact cobordism. Assume that there is a handle decomposition with at most one 0-handle, and denote the handle bodies by $M_i, i \in \{0,\...

2
votes

1
answer

189
views

When reading chaptes 7 of Akbulut's book about $4-$manifolds, he describes a handle decomposition for a manifold given a Lefschetz fibration over $S^2$. The idea is to extend the preimage of a disk ...

7
votes

1
answer

1k
views

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 $...

13
votes

2
answers

582
views

It's easy to derive a presentation of the fundamental group of a 4-manifold if you have a Kirby diagram: The 1-handles are generators and the 2-handles are the relations. The 3- and 4-handles, which ...

14
votes

2
answers

955
views

I'm an undergraduate student, interested in the low dimensional topology, in particular, the 4-manifold theory.
I have a question.
In the knot theory, the Reidemeister moves play fundamental roles.
...

17
votes

1
answer

2k
views

A celebrated theorem of Rohlin states the following
An oriented closed 4-manifold $M^4$ bounds an oriented 5-manifold if and only if the signature of $M^4$ is zero.
Simple homological arguments ...

55
votes

3
answers

5k
views

Every orientable 3-manifold can be obtained from the 3-sphere by doing surgery along a framed link. Kirby's theorem says that the surgery along two framed links gives homeomorphic manifolds if and ...