Mathematics Colloquia and Seminars

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It is a fact that a Morse function on a manifold induces a handle decomposition for that manifold. It is also a fact that any 3-manifold can be realized as the boundary of a 4-manifold constructed as a single 4-ball and a collection of four-dimensional 2-handles attached to its 3-sphere boundary. This creates a framed link in the 3-sphere that describes a particular 3-manifold. Kirby then defined two operations on framed links in the 3-sphere to show that two 3-manifolds are diffeomorphic if and only if the framed links used to describe them are related through a sequence of these operations. These moves are now referred to as the "Kirby Calculus".

The goal of this talk is to outline how to prove Kirby's theorem. We will review the facts mentioned above and then step through the Cerf theory argument that Kirby used in his seminal paper "A Calculus for Framed Links in S^3".