Decomposing manifolds into handlebodies
This is joint work with Stephan Tillmann and partly with Joel Hass and Mark Bell.
Smale's 1960s theory of handle decompositions of manifolds has proved a fundamental tool in topology, analysis, geometry and dynamical systems. But already in 1898 Heegaard had introduced the idea of splitting a 3-manifold into two simple handlebodies, now called a Heegaard splitting. These splittings have been used to define several important invariants of 3-manifolds, including the Casson invariant, which counts representations of the fundamental group into \(SU(2)\), and Heegaard Floer homology of Ozsvath and Szabo.
Around 2012 Gay and Kirby introduced a method of trisecting 4-manifolds into three simple handlebodies, arising from their work on generalisations of Lefschetz fibrations and Morse 2-functions. We show that there is a natural way of decomposing an n-manifold into \(k +1\) simple handlebodies, where \(n=2k\) or \(2k+1\), which we call a multisection. For \(n=3 \) this gives Heegaard splittings and \(n=4\) gives Gay Kirby trisections. Properties and potential applications of these decompositions will be discussed. In dimension 4 we have implemented an algorithm to construct trisections of 4-manifolds using our approach and this has been applied to some standard examples such as the \(K3\) surface to compute the simplest possible trisection.
Professor Hyam Rubinstein, University of Melbourne