Geometric homology revisited

Given a cohomology theory, there is a well-known abstract way to define the dual homology theory using the theory of spectra. In [4] the author provides a more geometric construction of the homology theory, using a generalization of the bordism groups. Such a generalization involves in its definition the vector bundle modification, which is a particular case of the Gysin map. In this paper we provide a more natural variant of that construction, which replaces the vector bundle modification with the Gysin map itself, which is the natural push-forward in cohomology. We prove that the two constructions are equivalent.