  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.
