Fold cobordisms and a Poincaré–Hopf-type theorem for the signature

Abstract

We give complete geometric invariants of cobordisms of framed fold maps. These invariants consist of two types. We take the immersion of the fold singular set into the target manifold together with information about non-triviality of the normal bundle of the singular set in the source manifold. These invariants were introduced in the author's earlier works. Secondly we take the induced stable partial framing on the source manifold whose cobordisms were studied in general by Koschorke. We show that these invariants describe completely the cobordism groups of framed fold maps into R^n. Then we are looking for dependencies between these invariants and we study fold maps of 4k-dimensional manifolds into R^2. We construct special fold maps which are representatives of the fold cobordism classes and we also compute cobordism groups. We obtain a Poincare-Hopf type formula, which connects local data of the singularities of a fold map of an oriented 4k-dimensional manifold M to the signature of M. We also study the unoriented case analogously and prove a similar formula about the twisting of the normal bundle of the fold singular set.