A note on Thom classes in general cohomology

This note is dedicated to the second author’s teacher, Professor Atsuo Komatsu, in celebration of his seventieth birthday.It is well known [4], [7] that the theory of characteristic classes in general cohomology is essentially based upon one theorem, the Leray-Hirsch Theorem. We further claim [8] that the entire theory could be developed merely from the Künneth Formula if under suitable conditions a truly relative version of the Meyer-Vietoris sequence exists in the general cohomology theory. In his lectures delivered at Aarhus in 1968, Dold [6] set up the necessary machinery including the Leray-Hirsch Theorem to define Chern Classes with values in general cohomology. However, he then stated [6, p. 47] that he “found a difficulty here in choosing adequate orientations (Thorn Classes) for the bundles involved”, and proceeded differently, discarding the “classical” approach used in both ordinary cohomology [9] and K-theory [3]. Later, he [7] published a more categorical work, although the approach to Chern Classes was basically unchanged from his previous work. Consequently, the possibility of the classical direct approach to the theory has remained open.