The Rost invariant has trivial kernel for
quasi-split groups of low rank
For G an almost simple simply connected algebraic group defined over a field
F, Rost has shown that there exists a canonical map
RG: H1(F, G) --> H3(F, Q/Z(2)). This
includes the Arason invariant for quadratic forms and Rost's mod 3 invariant
for Albert algebras as special cases. We show that RG has trivial
kernel if G is quasi-split of type E6 or E7. A
case-by-case analysis shows that it has trivial kernel whenever G is
quasi-split of low rank. This has a direct application to structurable
algebras of dimension 56 related to Albert algebras: Such an algebra which
is isotopic to a quasi-split one is actually quasi-split.
(This paper has appeared in Commentarii Mathematici Helvetici 76 #4 (2001),
684-711)
R. Skip Garibaldi <skip@member.ams.org>