math.GN — General Topology

Smooth functions with isolated critical values and systems of fibers

Contributed by Naok Kitazawa

In the 2020s, Marzantowicz and Michalak have introduced the notion of system of smooth hypersurfaces and discussed Morse functions compatible with them: the hypersurfaces are regular level sets of the functions. This is a kind of fundamental theory on Morse functions and related differential topology and geometric topology. We extend this to a certain case: we discuss systems of regular level sets and critical level sets (regular fibers and singular fibers) of smooth functions with isolated critical values.