  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.
