multisection

Multisection is a concept in geometric topology that provides a way to decompose a closed smooth manifold into several simpler pieces, generalizing both Heegaard splittings of 3-manifolds and trisections of 4-manifolds. It was introduced by Gay and Kirby as a higher-dimensional extension of trisections, offering a uniform framework to study smooth manifolds via combinatorial data.

In a multisection, a closed n-manifold M is written as a union of k submanifolds X1, X2, ..., Xk that satisfy a set of regularity conditions. Each Xi is a standard, “one-handle”–type piece, analogous to a neighborhood built by attaching 1-handles to a ball. The intersections of these pieces have controlled, lower-dimensional structure: pairwise intersections Xi ∩ Xj are simpler submanifolds, and the pattern continues for higher-order intersections, with a central common intersection X1 ∩ X2 ∩ ... ∩ Xk that encapsulates the core topology. The precise models of these intersections depend on the dimension and the chosen order k, but the general aim is to have all pieces and their overlaps be understood and manageable.

A key feature of multisections is the associated multisection diagram, which encodes the manifold’s data in terms of the common intersection and the way the Xi glue together. In dimension three, a multisection with k = 2 recovers a Heegaard splitting; in dimension four, k = 3 yields a trisection. For higher dimensions, multisections provide a scalable framework for constructing, classifying, and studying smooth manifolds through combinatorial diagrams.

Applications include simplifying handle decompositions, analyzing mapping class groups, and guiding the construction of invariants via diagrammatic methods.