  This article proves the existence of a bound on the sum of local Betti
numbers of a real analytic germ by a polynomial function of its multiplicity.
This result can be interpreted as a localization of the classical
Oleinik-Petrovsky bound (also known as Thom-Milnor bound) on the sum of Betti
numbers of a semi-algebraic set.
  The proof relies on an interplay between geometric and algebraic arguments
whose key elements are the tangent cone of the germ, the Thom-Mather
topological trivialization theorem, the Oleinik-Petrovsky bound, and a result
by D. Mumford and J. Heintz bounding the degrees of the generators of an ideal
by a polynomial function of the geometric degree of its associated variety.
  Our result is then applied to yield bounds on invariants from singularity
theory, such as the Lipschitz-Killing curvature invariants and the Vitushkin
variations (which include the local density of a germ).
