Computing trisections of 4-manifolds
arXiv:1711.02763 · doi:10.1073/pnas.1717173115
Abstract
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 -dimensional manifold as a union of three -dimensional handlebodies. The complexity of the -manifold is captured in a collection of curves on a surface, which guide the gluing of the handelbodies. The algorithm begins with a description of a manifold as a union of pentachora, or -dimensional simplices. It transforms this description into a trisection. This results in the first explicit complexity bounds for the trisection genus of a -manifold in terms of the number of pentachora (-simplices) in a triangulation.
15 pages, 9 figures