Let us mention that in later sections we will define the group B.W I J / for an arbitrary (not necessarily closed) manifold W . At the moment, however, we consider only maps of closed manifolds, just to simplify the exposition.Carefully choosing the set J of singularities, one may derive cobordism groups related to various objects in geometry and topology. Example 1.1 Since a proper submersion is a smooth fiber bundle, the cobordism group of submersions is closely related to diffeomorphism groups of smooth manifolds. It is also known to be related to various infinite loop spaces, moduli spaces of Riemann surfaces, the cobordism category as in Galatius et al [17] A priori J -cobordism groups do not form generalized cohomology theories since, for example, J -cobordism groups are not defined for topological spaces. In the current paper we propose a counterpart of B.W I J / that for a wide range of sets J can be used to compute B.W I J / in the same way as singular cohomology groups H n .W I R/ can be used to compute De Rham cohomology groups H