On a theorem of A. Borel

Abstract

trivially on the fibre F. In what follows, cohomology will always mean singular (cubical) cohomology; the standard notation, C*, Z*, etc., for cochains, cocycles, etc., will be used, and coefficients will always be taken in a commutative ring K with a unit element. Throughout, it will be assumed that the cohomology algebra H*(F; K) is a graded tensor product of monogenic, possibly truncated, polynomial algebras with transgressive generators of positive degrees, and that H*(F; K) is of finite type (i.e., has a finite (additive) K-basis in each dimension) with IGH°(F; K). My aim here is, as in [3], to study the relation between the cohomology groups of E and those of B and F. I recall that in [3] details were given of the method of Hirsch in singular cohomology theory, in which under suitable conditions, a coboundary operator d is proved to exist on the tensor product C*(B; K)®KH*(F; K), together with a map u:C*(B; K)®KH*(F; K) —>C*(E; K) which induces isomorphisms on cohomology groups. The results proved here below show that, in fact, under the conditions given above, d and u, in the Hirsch method, can be defined by explicit formulae in terms of cup- and cup-l-products obtained from the generators of H*(F; K). These explicit formulae of course generalize the explicit formulae given in [3] up to dimension 2m — 1 for d and u, in case F is a space of type X(tr, n). Collected together, the results may be regarded as a weak generalization of a theorem of A. BoreK1), referred to in the title. I use the adjective weak because whereas Borel proves the existence of a multiplicative isomorphism