Decomposing manifolds into handlebodies


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
    Professor Hyam Rubinstein, University of Melbourne