A classifying space \(BG\) for an algebraic group \(G\) over a field \(F\) is the quotient variety \(\text{SL}_n/G\) for an embedding \(\rho:G\hookrightarrow \text{SL}_n\) over \(F\). The unramified cohomology of \(BG\) is defined as follows. Let \(\text{Gal}(F)\) denote the absolute Galois group of \(F\), and for \(d\geq 2\) let \(({\mathbb Q}/{\mathbb Z})'(d-1)=\overrightarrow{\lim}_n {\mu}_n^{\otimes (d-1)}\) with \(n\) not divisible by the characteristic of \(F\), and denote by \(H^d(F)\) the Galois cohomology group \(H^d(\text{Gal}(F),({\mathbb Q}/{\mathbb Z})'(d-1))\). The unramified cohomology \(H^d_{\text{nr}}(BG/F)\) is then defined to be the intersection of the kernels of the residue homomorphisms \[ \partial_v:H^d(F(BG))\to H^{d-1}(F(v))\;, \] where \(v\) ranges over all discrete valuations of the function field \(F(BG)\) over \(F\) and \(F(v)\) denotes the residue field at such a discrete valuation \(v\). Since the natural homomorphism \(H^d(F)\to H^d_{\text{nr}}(BG/F)\) is split by evaluation at the distinguished point of \(BG\), \(H^d(F)\) splits off as a direct summand of \(H^d_{\text{nr}}(BG/F)\) with its complement denoted by \(H^d_{\text{nr}}(BG/F)_{\text{norm}}\). This latter group only depends on \(G\) and \(F\).
The aim of this paper is to complete the computation of \(H^3_{\text{nr}}(BG/F)_{\text{norm}}\) for semisimple simply connected algebraic groups \(G\) over arbitrary fields \(F\). The computation can be reduced to the case of \(G\) being simple. In [Ann. Sci. Éc. Norm. Supér. (4) 35, No. 3, 445–476 (2002; Zbl 1022.14015)], A. Merkurjev computed \(H^3_{\text{nr}}(BG/F)_{\text{norm}}\) for all simple and classical such \(G\), and the present paper deals with the remaining cases where \(G\) is exceptional. The main theorem states that \(H^3_{\text{nr}}(BG/F)_{\text{norm}}=0\) for any exceptional simple simply connected algebraic group except in the case where \(\text{char}(F)\neq 2\) and \(G\) is of type \(^3D_4\) and has a nontrivial Tits algebra, in which case \(H^3_{\text{nr}}(BG/F)_{\text{norm}}={\mathbb Z}/2{\mathbb Z}\). As a corollary which is of considerable interest in its own right, one obtains that if \(G\) is simply connected of type \(^3D_4\) or \(^6D_4\) over a field of characteristic not \(2\), and if \(G\) has a nontrivial Tits algebra, then \(BG\) is not stably rational, thus providing new additions to Merkurjev’s list of examples with \(BG\) not stably rational.
An important ingredient in the proof of the main theorem is the fact that every isotropic trialitarian group embeds in a group of type \(^2E_6\) with trivial Tits algebras. This is proved using Galois descent and interpretations of exceptional groups as acting on nonassociative algebras.