 The trivialization is constructed from smooth paths in "U", and it can be shown that the homotopy class of the diffeomorphism depends only on the choice of a homotopy class of paths from "b" to 0.In particular, if "U" is contractible, there is a well-defined diffeomorphism up to homotopy.The diffeomorphism from "X""b" to "X"0 induces an isomorphism of cohomology groups
and since homotopic maps induce identical maps on cohomology, this isomorphism depends only on the homotopy class of the path from "b" to 0.