# Questions tagged [kirby-calculus]

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

12 questions
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### Surgery (Kirby) diagram of connected sum of lens spaces?

What is the kirby/surgery diagram for a connected sum of two lens spaces? Question 1: is it just the unlink with $p/q$ coefficients? Question 2: if not, what manifold is the unlink with $p_i/q_i$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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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### 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 ...
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### 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 (...
360 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,\... 1answer 122 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 ... 1answer 920 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$...
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 ...