Geometric interpretation of First Betti numbers of smooth functions orbits

Abstract

Let $M$ be a 2-disk or a cylinder, and $f$ be a smooth function on $M$ with constant values at $\partial M$, devoid of critical points in $\partial M$, and exhibiting a property wherein for every critical point $z$ of $f$ there is a local presentation of $f$ near $z$ that is a homogeneous polynomial without multiple factors. We consider $V$ to be either the boundary $\partial M$ (in the case of a 2-disk) or one of its boundary components (in the case of a cylinder) and $\mathcal{S}^{'}(f, V)$ to consist of diffeomorphisms preserving $f$, isotopic to the identity relative to $V$. We establish a correspondence: the first Betti number of the $f$-orbit is shown to be equal to the number of orbits resulting from the action of $\mathcal{S}^{'}(f,V)$ on the internal edges of the Kronrod-Reeb graph associated with $f$.