Stable parallelizability of partially oriented flag manifolds

Parameswaran Sankaran and Peter Zvengrowski

Vol. 128 (1987), No. 2, 349–359

DOI: 10.2140/pjm.1987.128.349

Abstract

This paper solves the questions of stable parallelizability and parallelizability for the family of partially oriented (p.o.) flag manifolds, except for a few undecided cases. In particular, for the oriented Grassmannians G_k(Rⁿ) it is proved that apart from the spheres S¹, S³, and S⁷ only G₃(R⁶) is parallelizable, and only G₂(R⁴) is stably parallelizable and not parallelizable. Negative results are derived for the most part using KO theory and the “inclusion method”, while positive results are mainly based on the “λ² construction”.

Mathematical Subject Classification 2000

Primary: 57R15

Milestones

Received: 2 January 1986

Published: 1 June 1987