Vector bundles over (8k + 3)-dimensional manifolds

Tze-Beng Ng

Vol. 121 (1986), No. 2, 427–443

DOI: 10.2140/pjm.1986.121.427

Abstract

Let M be a closed, connected, smooth and 3-connected mod 2 (i.e. H_i(M;Z₂) = 0, 0 < i ≤ 3) manifold of dimension 3 + 8k with k > 1. We obtain some necessary and sufficient condition for the span of a (3 + 8k)-plane bundle η over M to be greater than or equal to 7 or 8. We obtain, for M 4-connected mod 2 and satisfying Sq² Sq¹Hⁿ⁻⁸(M) = Sq²Hⁿ⁻⁷(M), where n = dimM ≡ 11 mod 16 with n > 11, that span M ≥ 8 if and only if χ₂(M) = 0. Some applications to product manifolds and immersion are given.

Mathematical Subject Classification 2000

Primary: 57R25

Secondary: 55S45

Milestones

Received: 8 June 1984

Published: 1 February 1986