H3and Generic Matrices

Unramified degree three invariants of reductive groups

“…F. Bogomolov proved in [3,Lemma 5.7] (see also [2,Theorem 5.10]) that connected groups have no non-trivial degree 2 unramified invariants. In [15] and [16], D. Saltman proved that the projective linear group PGL n has no non-trivial degree 3 unramified invariants.…”

Degree 3 unramified cohomology of classifying spaces for exceptional groups

“…In [3], Bogomolov proved that H 1 nr (E) = H 2 nr (E) = 0 for any connected group G. For d = 3, the triviality of the group H 3 nr (E) has been verified in several cases. For G = PGL n (projective general linear group), the triviality of the group was proved by Saltman in [18]. For a simple simply connected group G, the same result was proved by Merkurjev (classical groups) [14] and Garibaldi (exceptional groups) [5].…”

Degree three invariants for semisimple groups of types $B$, $C$, and $D$

“…for all p and i = i p , where the sums range over all l such that w i,l ∈ W ′ , the first equation in ( 21) is equivalent to the following equation (23) f p (T 1 , . .…”

“…In [5], Bogomolov showed that connected groups have no nontrivial degree 2 unramified invariants, i.e., Inv 2 nr (G) = 0 for a connected group G. In [22] and [23], Saltman showed that the group Inv 3 nr (PGL n ) is trivial. Recently, Merkurjev has shown that the group Inv 3 nr (G) is trivial if G is a split simple group [18] or a split semisimple group of type A [14] over an algebraically field F of characteristic 0.…”