Self-intersection class for singularities and its application to fold maps

Abstract

Let f : M → N f :M \to N be a generic smooth map with corank one singularities between manifolds, and let S ( f ) S(f) be the singular point set of f f . We define the self-intersection class I ( S ( f ) ) ∈ H ∗ ( M ; Z ) I(S(f)) \in H^*(M; \mathbf {Z}) of S ( f ) S(f) using an incident class introduced by Rimányi but with twisted coefficients, and give a formula for I ( S ( f ) ) I(S(f)) in terms of characteristic classes of the manifolds. We then apply the formula to the existence problem of fold maps.