Bordism of regularly defective maps

To a topological space V we assign the bordism group N-n(def) (V) on closed n-dimensional manifolds M. These are triples (M,Delta,f) where Delta is a closed submanifold Deltasubset ofM and f a continuous map f: M\Delta-->V. We briefly review the construction of the defect complex DV given by M. Rost in [17] and show that N-n(def) (V) is isomorphic to ordinary bordism N-n(DV). The bordism classes in N-n(def)(V) congruent to Nn(DV) are detected by characteristic numbers twisted with cohomology classes of DV. Some of these numbers can be described without reference to the defect complex. As an example we treat the case of the circle V=S-1. We compute N-n(def) (S-1), construct a basis and a complete set of characteristic numbers.