Arnold's Seminar
7th December, 1999

Fabien NAPOLITANO (Paris-9)
"Rigidity and stability principles in singularity theory"

We present two general principles in relation with the topology of
the complement to a discriminant set $\Xi$ in affine space $F$.

The rigidity principle assert that we can in many situations replace $F$ by
a much smaller affine subspace $E$ without changing much the topology of the
complement to the discriminant (the cohomology rings of $F \setminus \discr$
and $E \setminus \discr$ are isomorphic for instance).  For example it implies
Smale-Hirsh principle (see Vassiliev complements of discriminants os smooth
maps).

The stability principle gives conditions on a sequence
$(F_k)$ of affine finite dimensionnal subspaces of $F$ implying that the
cohomology rings of the sequence $F_k \setminus \discr$ stabilize as $k$ tends
to infinity.  For example it implies the stabilisation of the cohomology groups
of the sequence $\operatorname{Br} n$ of braid groups as $n$ tends to infinity
(see Arnold topological invariants of algebraic functions).

These two theorems are based on characteristic numbers defined for the strata
of the discriminant (roughly speaking they apply when the characteristic
numbers are greater than $1$).