# Decomposing manifolds into handlebodies

## Seminar/Forum

107
Peter Hall
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.