Computing trisections of 4-manifolds

Significance Algorithms that decompose a manifold into simple pieces reveal the geometric and topological structure of the manifold, showing how complicated structures are constructed from simple building blocks. This note describes a way to algorithmically construct a trisection, which describes a four-dimensional manifold as a union of three four-dimensional 1-handlebodies. The complexity of the 4-manifold is captured in a collection of curves on a surface, which guide the gluing of the 1-handlebodies. The algorithm begins with a description of a manifold as a union of pentachora or four-dimensional simplices. It transforms this description into a trisection.