4-fields on (4k + 2)-dimensional manifolds

Tze-Beng Ng

Vol. 129 (1987), No. 2, 337–348

DOI: 10.2140/pjm.1987.129.337

Abstract

Let M be a closed, connected, smooth and 2-connected mod 2 (i.e., H_i(M,Z₂) = 0, 0 < i ≤ 2) manifold of dimension n = 4k + 2 with k > 1. We obtain some necessary and sufficient conditions for the span of an n-plane bundle η over M to be greater than or equal to 4. For instance for k odd span M ≥ 4 if and only if χ(M) = 0. Some applications to immersion are given. In particular if n = 2 + 2^l, l ≥ 3 and w₄(M) = 0 then M immerses in R²ⁿ⁻⁴.

Mathematical Subject Classification 2000

Primary: 57R25

Secondary: 55S45, 57R19, 57R42

Milestones

Received: 24 March 1986

Published: 1 October 1987