  Let M be a compact smooth manifold of dimension n with or without boundary,
and f : M $\rightarrow$ R be a smooth Gaussian random field. It is very natural
to suppose that for a large positive real u, the random excursion set {f $\ge$
u} is mostly composed of a union of disjoint topological n-balls. Using the
constructive part of (stratified) Morse theory we prove that in average, this
intuition is true, and provide for large u the asymptotic of the expected
number of such balls, and so of connected components of {f $\ge$ u}, see
Theorem 1.2. We similarly show that in average, the high nodal sets {f = u} are
mostly composed of spheres, with the same asymptotic than the one for excursion
set. A refinement of these results using the average of the Euler
characteristic given by [2] provides a striking asymptotic of the constant
defined by F. Nazarov and M. Sodin, again for large u, see Theorem 1.11. This
new Morse theoretical approach of random topology also applies to spherical
spin glasses with large dimension, see Theorem 1.14.
