  Given an $S^1$-manifold with isolated fixed points, some recent papers are
concerned with the relationship between the least number of fixed points and
the characteristic numbers of this manifold, and their proofs have some similar
features. The main purpose of this short survey article is, by using the
language of equivariant cohomology, to present a unified method to deal with
such problems, of which the related known results are direct corollaries.
