Questions tagged [kirby-calculus]

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

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1answer
284 views

Presentations of exotic 4-manifolds

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, ...
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0answers
129 views

Mapping class group and surgery theory

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 $\...
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0answers
169 views

Kirby's theorem for 4-manifolds

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
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1answer
655 views

Inverse Kirby knot

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
1answer
48 views

Normal form of framed links under Kirby moves

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
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0answers
103 views

The Kirby diagram of a manifold glued along the lens space $L(p,1)$

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
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1answer
373 views

Akbulut's cork involution

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
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2answers
972 views

Cobordism and Kirby calculus

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
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1answer
303 views

Kirby calculus on Mazur manifolds

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.
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0answers
185 views

Tubular neighbourhoods are unique up to ambient isotopy?

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
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1answer
226 views

Obtaining the bounding 4-manifold from the Heegaard diagram

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
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1answer
251 views

Framings for 2-surgeries on 4-manifolds

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
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0answers
98 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.
3
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0answers
94 views

What's a completely computational/syntactical model for handle decompositions of manifolds?

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
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0answers
208 views

Cap product for (co)homology from handle decompositions/Kirby diagrams

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
2answers
537 views

Are there Kirby diagrams with 3-handles?

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
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1answer
176 views

Lefschetz Fibrations and disk bundles

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
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1answer
1k 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 $...
13
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2answers
526 views

Given a Kirby diagram of a 4-manifold, what's its homotopy 2-type?

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 ...
13
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2answers
902 views

Construction of invariants of 4-manifolds with the Kirby calculus

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
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1answer
2k views

Proofs of Rohlin's theorem (an oriented 4-manifold with zero signature bounds a 5-manifold)

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 ...
53
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3answers
5k views

Kirby calculus and local moves

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 ...