We give complete geometric invariants of cobordisms of framed fold maps. These
invariants are of two types. We take the immersion of the fold singular set into the
target manifold together with information about nontriviality of the normal bundle of
the singular set in the source manifold. These invariants were introduced in the
author’s earlier works. 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 .
Then we look for dependencies between these invariants and we study fold maps of
–dimensional
manifolds into
.
We construct special fold maps, which are representatives of the fold cobordism
classes and we also compute the cobordism groups. We obtain a Poincaré–Hopf-type
formula, which connects local data of the singularities of a fold map of an oriented
–dimensional
manifold
to the
signature of
.
We also study the unoriented case analogously and prove a similar formula about the
twisting of the normal bundle of the fold singular set.
