Galois cohomology of quasi-split groups over fields of cohomological dimension <= 2

Let k be a perfect field with cohomological dimension less than or equal to 2. Serre's conjecture II claims that the Galois cohomology set H-1(k,G) is trivial for any simply connected semi-simple algebraic G/k and this conjec ture is known for groups of type (1)A(n) after Merkurjev-Suslin and for cla ssical groups and groups of type F-4 and G(2) after Bayer-Parimala. For any maximal torus T of G/k, we study the map H-1(k, T) --> H-1(k, G) using an induction process on the type of the groups, and it yields conjecture II fo r all quasi-split simply connected absolutely almost k-simple groups with t ype distinct from E-8. We also have partial results for E-8 and for some tw isted forms of simply connected quasi-split groups. In particular, this met hod gives a new proof of Hasse principle for quasi-split groups over number fields including the E-8-case, which is based on the Galois cohomology of maximal tori of such groups.