 
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
hcobordism theorem. Nowadays some authors call 4dimensional 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 3manifold 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 3manifolds as boundaries of the
4ball with 2handles. My own interest has been dealing with 4manifold
handlebodies to solve 4manifold problems, where the interactions of 1 and
2handles play particularly crucial role (e.g. my paper "On 2dimensional
homology classes of 4manifolds" 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
