Classification of Singularities. Part I: The Thom-Boardman Invariants

Abstract

For a mapping f: X → Y we can make the following rudimentary classification of singularities. We say that f has a singularity of type S _r at x in X if (df) _x drops rank by r; i.e., if rank (df) _x = min (dim X, dim Y) - r. Denote by S _r (f) the singularities of f of type S _r . Recall that in the proof of the Whitney Immersion Theorem we introduced the submanifolds S _r of J ¹ (X, Y) consisting of jets of corank r. (See II, Theorem 5.4.) Clearly S _r (f) = (j¹f)^-1(S _r ). To prove the Whitney Theorem we showed that if X and Y have the “right” relative dimensions then generically S _r (f) = ∅ (r > 0) and f has no singularities; i.e., f is an immersion. Without restricting the relative dimensions of X and Y we can still say that generically S _r (f) is a submanifold of X and codim S _r (f) = codim S _r = r² + er where e = |dim X - dim Y|. This statement follows immediately from the Thorn Transversality Theorem and II, Theorem 4.4. In particular, the set of mappings for which j¹f⋔S _r (for all r) is residual. Besides the Transversality Theorem, the main fact used in the proof of this statement is that S _r is actually a submanifold of J¹(X, Y). We shall sketch a different proof of this fact in order to motivate the material in §4.