The goal of this note is to define the biparametric persistence diagrams for smooth generic mappings \(h=(f,g):M\to\Real^2\) for smooth compact manifold \(M\). Existing approaches to multivariate persistence are mostly centered on the workaround of absence of reasonable algebraic theories for quiver representations for lattices of rank 2 or higher, or similar artificial obstacles.
Singularities of mappings into the plane
We will rely on the standard facts about generic smooth mappings into two-dimensional manifolds: for such mappings, the set of critical points is a smooth curve \(\Sigma\) in \(M\), which is immersed outside of a finite number of pleats: near generic point of \(\Sigma\), there are local coordinates on \(M\) in which the mapping is locally given by
(folds), and near isolated points of the curve of critical points, in some coordinates the mapping is given by