Kirby calculus? Handlebodies on smooth manifolds were Smale's way of looking at the Summary: Kirby calculus? Handlebodies on smooth manifolds were Smale's way of looking at the Morse theory; he used them beautifully in his proof of the high dimensional h-cobordism theorem. Nowadays some authors call 4-dimensional handle- bodies1 "Kirby diagrams" and handle slides "Kirby calculus" In his 1974 seminal paper "A calculus of framed links in S3 " (which was published in 1978 in "Invent. Math") Kirby showed how two handlebodies describing the same 3-manifold have to be related to each other, by giving a handle interpretation of the Cerf theory, which describes how two Morse functions on the same manifold are related to each other (births, deaths, handle slides). Though this theorem does not have a useful corollary by itself, it gives a concrete way of looking at 3-manifolds as boundaries of the 4-ball with 2-handles. My own interest has been dealing with 4-manifold handlebodies to solve 4-manifold problems, where the interactions of 1- and 2-handles play particularly crucial role (e.g. my paper "On 2-dimensional homology classes of 4-manifolds" in "Math Proc. Camb. Phil. Soc." (1977)) Merely describing a handlebody of a manifold does not prove you a theorem, but it can be a first step towards solving a hard problem (usually Collections: Mathematics