\(\def\Real{\mathbb{R}}

\def\phd{\mathtt{PH}}

\def\CAT{\mathtt{CAT}}

\)

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

\[

y_1=x_1, y_2=q(x_2,\ldots,x_m)

\]

(folds), and near isolated points of the curve of critical points, in some coordinates the mapping is given by

\[

y_1=x_1, y_2=x_2^3+x_1x_2+q(x_3,\ldots,x_m),

\]

(pleats).…