1. Introduction. In
this paper, we show that the set of all C∞ Thom-Boardman maps from
an n-dimensional manifold is not open iff corank two singularities occur
generically. The latter is known to occur iff either n ≦ p and 2p ≦ 3n − 4 or
n > p and 2p ≧ n + 4. In the course of the proof, we establish a variation of
Mather’s Multitransversality Theorem: we show that jets have extensions which
are multitransverse to given submanifolds of the jet bundle except possibly
at the original jet. As an application of this extension theorem, we show
that, in Mather’s “nice range of dimensions,” each iet z has a representative
f(z = jkf(x)) such that f is infinitesimally stable on a deleted neighborhood of
x.
