On the Product of Sphere Bundles and the Duality Theorem Modulo Two

Abstract

Given two sphere bundles (5 and 52 over the base complexes K1 and 1K2 respectively, it is possible to define in a natural way a over the product complex K1 X K2 . When K1 = K2= K (say), the part of the product bundle over the diagonal of the product complex K X K is the product bundle in the sense of Whitney.' We shall prove in the present paper that a certain duality theorem holds for the product bundle over K1 X K2 and that Whitney's duality theorem for sphere bundles follows from this more general duality theorem as a consequence. (Throughout the paper coefficients mod 2 will be used.) The idea of this proof seems to be quite different from Whitney's original one, of which only a brief sketch is known.2 The paper is divided into three sections. In ?1 some preliminary considerations and theorems on vector fields are given. A duality theorem for the product bundle over K, X K2 is then proved in ?2. ?3 is devoted to a proof of Whitney's duality theorem.