The Allison–Faulkner construction of

We show that the Tits index $E_{8,1}^{133}$ cannot be obtained by means of the Tits construction over a field with no odd degree extensions. The proof uses a general method coming from the theory of symmetric spaces. We construct two cohomological invariants, in degrees $6$ and $8$ , of the Tits construction and the more symmetric Allison–Faulkner construction of Lie algebras of type $E_8$ and show that these invariants can be used to detect the isotropy rank.

Cohomological Invariants in Positive Characteristic

We determine the mod $p$ cohomological invariants for several affine group schemes $G$ in characteristic $p$. These are invariants of $G$-torsors with values in étale motivic cohomology, or equivalently in Kato’s version of Galois cohomology based on differential forms. In particular, we find the mod 2 cohomological invariants for the symmetric groups and the orthogonal groups in characteristic 2, which Serre computed in characteristic not 2. We also determine all operations on the mod $p$ étale motivic cohomology of fields, extending Vial’s computation of the operations on the mod $p$ Milnor K-theory of fields.

Brauer groups of moduli of hyperelliptic curves via cohomological invariants

Using the theory of cohomological invariants for algebraic stacks, we compute the Brauer group of the moduli stack of hyperelliptic curves ${\mathcal {H}}_g$ over any field of characteristic $0$ . In positive characteristic, we compute the part of the Brauer group whose order is prime to the characteristic of the base field.

Cohomological invariants of representations of 3-manifold groups

Suppose [Formula: see text] is a discrete group, and [Formula: see text], with [Formula: see text] an abelian group. Given a representation [Formula: see text], with [Formula: see text] a closed 3-manifold, put [Formula: see text], where [Formula: see text] is a continuous map inducing [Formula: see text] which is unique up to homotopy, and [Formula: see text] is the pairing. We extend the definition of [Formula: see text] to manifolds with corners, and establish a gluing law. Based on these, we present a practical method for computing [Formula: see text] when [Formula: see text] is given by a surgery along a link [Formula: see text]. In particular, the Chern–Simons invariant can be computed this way.