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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

  

Source: Akbulut, Selman - Department of Mathematics, Michigan State University

 

Collections: Mathematics