Nine-fields on manifolds

Abstract

E. Thomas in 1969 in a survey paper "Vector Fields on Manifolds" (loc. cit.) made several conjectures about the span of manifolds. We solve Problem 5 of that paper, namely, that when n is congruent to 31 mod 32 and Mn is 7-connected, then Mn admits a tangent 9-field if, and only if, the mod 2 semi-Kervaire characteristic X2(M) vanishes. Indeed we obtain the same conclusion by requiring Mn to be only 7-connected mod 2 instead of 7-connected. We obtain necessary and sufficient conditions for an n-plane bundle ξ over Mn whose dimension n is congruent to 7 mod 8 with n > 15 to have span≥9. The results are applied to the tangent bundle of Mn and we prove one of Thomas's conjectures concerning the existence of 9-fields on such an n-dimensional manifold Mn. The method of proof consists in realizing the obstructions to the existence of 9 linearly independent sections to ξ via secondary, tertiary and fourth-order cohomology operations on the Thom class of ξ and the evaluation of these operations on the Thom class of the tangent bundle of Mn. We also prove that when n is congruent to 23 mod 32, span Mn≥8 implies span Mn≥9. © 1991.