Nonsmooth manifold decompositions

TL;DR

This paper investigates how continuous selections of smooth functions induce stratifications on manifolds, especially focusing on the case of three functions on 4-manifolds, offering new insights into trisection theory.

Contribution

It introduces a framework linking smooth function selections to manifold stratifications and applies it to analyze trisections of 4-manifolds.

Findings

Stratifications are algebraically defined smooth submanifolds under generic conditions.

Nondegenerate critical points lead to a topological stratification.

Provides a new perspective on the theory of trisections in 4-manifolds.

Abstract

We study the structure induced on a smooth manifold by a continuous selection of smooth functions. In case such selection is suitably generic, it provides a stratification of the manifold, whose strata are algebraically defined smooth submanifolds. When the continuous selection has nondegenerate critical points, stratification descends to the local topological structure. We analyze this structure for the maximum of three smooth functions on a 4-manifold, which provides a new perspective on the theory of trisections.