J-approximation of complex projective spaces by lens spaces

I. Dibag

Vol. 191 (1999), No. 2, 223–242

DOI: 10.2140/pjm.1999.191.223

Abstract

In this paper we study the group J(L^k(n)) of stable fibre homotopy classes of vector bundles over the lens space, L^k(n) = S^2k+1∕ℤ_n where ℤ_n is the cyclic group of order n. We establish the fundamental exact sequences and hence find the order of J(L^k(n)). We define a number N_k and prove that the inclusion-map i : L^k(n) → P_k(ℂ) induces an isomorphism of J(P_k(ℂ)) with the subgroup of J(L^k(n)) generated by the powers of the realification of the Hopf-bundle iff n is divisible by N_k. This provides the discrete approximation to the continuous case.

Milestones

Received: 28 July 1997

Revised: 23 October 1998

Published: 1 December 1999