Questions tagged [kirby-calculus]

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

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12
votes
2answers
818 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
votes
1answer
234 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.
0
votes
0answers
93 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
votes
1answer
195 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
votes
1answer
208 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 ...
3
votes
0answers
86 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
votes
0answers
90 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
votes
0answers
156 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
454 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
votes
1answer
147 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
votes
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 $...
12
votes
2answers
451 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 ...
12
votes
2answers
831 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. ...
16
votes
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 ...
52
votes
3answers
4k 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 ...