Geometric Aspects of Currents and Distributions.

A routine computation with the spectral sequence of the fibering (G2, T, G2/T) confirms the result of Borel4 that H*(G2) has a cyclic group of order two as torsion group in dimensions 6 and 9. 6. Let K be a connected subgroup of maximal rank in G. The inclusions T c K c G induce a fiberingfk: GIT -G/K of GIT over GIK with fiber KIT. If Pal, . . . , Pt are the planes through o in 2' which belong to K, set X*-= xa.al . . . Xak in H*(B). Our imbedding f* of H*(G/T) H*(B) now defines a homomorphism Xk: H*(G/T) -) Z by Xk (U) = coefficient of Xk in the expansion of f*(u) in terms of the multilinear base in the xj's. By Poincar6 duality, Xk can be realized by a definite element k e H*(G/T) once GIT is oriented and H2m(G/T) thus identified with Z. PROPOSITION III. fk,* maps a fundamental cocycle of G/K onto k. COROLLARY. If k e H*(B) is not divisible by a prime p, then G/K has no p-torsion. This proposition follows from the fact that the image under fk* of the fundamental class of G/K is the Poincar6 dual (in GIT) of the fiber KIT. 1 A. Borel, "Sur la cohomologie des espaces fibrds principaux et des espaces homognes de groupes de Lie compacts," Ann. Math., 57, 115-207; 1953. 2 R. Bott, "On Torsion in Lie Groups," these PROCEEDINGS, 40, 586-588, 1954; A. Borel, "Kihlerian Coset Spaces of Semisimple Lie Groups," these PROCEEDINGS, 40, 1147-1151, 1954. 3 W. Gysin, "Zur Homologie-Theorie der Faserungen und Abbildungen von Mannigfaltigkeiten," Comm. Math. Helv., 14,61-122, 1941. 4 A. Borel, "Sur P'homologie et la cohomologie des groupes de Lie compacts connexes," Am. J. Math., 76, 273-342, 1954.