Bondar' O. P.

In terms of the Euler characteristic, we obtain the condition of existence of Bott functions on differentiable manifolds that have a set of critical points formed by connected homeomorphic submanifolds.

We introduce a topological invariant of a manifold. In terms of this invariant, we obtain an estimate for the generalized Lyustemik-Shnirel’man category of the manifold considered and an estimate for the minimal number of singular submanifolds of a function on this manifold.

We consider differentiable functions on a manifold such that the set of their critical points is a disconnected union of smooth submanifolds. A topological characteristic of the manifold is introduced; in terms of this characteristic, we estimate the least possible number of critical submanifolds of these functions.