The minimal number of critical points of a smooth function on a closed manifold and the ball category

TL;DR

This paper investigates the nature of isolated critical points of smooth functions on closed manifolds, proposing a conjecture about their neighborhoods and linking the minimal number of critical points to a topological invariant.

Contribution

It introduces a conjecture about cylindrical ball neighborhoods of critical points and proves it for several classes, connecting critical point theory with topological invariants.

Findings

Conjecture holds for cone-like, Cornea reasonable, and Rothe H hypothesis critical points.

Minimal critical points relate to Singhof-Takens fillings in manifolds of dimension ≥ 6.

Existence of exotic critical points would imply deviations from the conjecture.

Abstract

Introduced by Seifert and Threlfall, cylindrical neighborhoods of isolated critical points of smooth functions is an essential tool in the Lusternik- Schnirelmann theory. We conjecture that every isolated critical point of a smooth function admits a cylindrical ball neighborhood. We show that the conjecture is true for cone-like critical points, Cornea reasonable critical points, and critical points that satisfy the Rothe H hypothesis. In particular, the conjecture holds true at least for those critical points that are not infinitely degenerate. If, contrary to the assertion of the conjecture, there are isolated critical points that do not admit cylindrical ball neighborhoods, then we say that such critical points are exotic. We prove a Lusternik-Schnirelmann type theorem asserting that the minimal number of critical points of smooth functions without exotic critical points on a closed manifold of dimension at least 6 is the same as the minimal number of elements in a Singhof-Takens filling of M by smooth balls with corners.