J-approximation of complex projective spaces by lens spaces

Abstract

In this paper we study the group J(Lk(n)) of stable fibre homotopy classes of vector bundles over the lens space, Lk(n) = S2k+1/ℤn where ℤn is the cyclic group of order n. We establish the fundamental exact sequences and hence find the order of J(Lk(n)). We define a number Nk and prove that the inclusion-map i : Lk(n) → Pk(ℂ) induces an isomorphism of J(Pk(ℂ)) with the subgroup of J(Lk(n)) generated by the powers of the realification of the Hopf-bundle iff n is divisible by Nk. This provides the discrete approximation to the continuous case.