# Cohomological Invariants

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This volume concerns invariants of $G$-torsors with values in mod $p$ Galois cohomology--in the sense of Serre's lectures in the book Cohomological invariants in Galois cohomology--for various simple algebraic groups $G$ and primes $p$. The author determines the invariants for the exceptional groups $F_4$ mod 3, simply connected $E_6$ mod 3, $E_7$ mod 3, and $E_8$ mod 5. He also determines the invariants of $\mathrm{Spin}_n$ mod 2 for $n \leq 12$ and constructs some invariants of $\mathrm{Spin}_{14}$. Along the way, the author proves that certain maps in nonabelian cohomology are surjective. These surjectivities give as corollaries Pfister's results on 10- and 12-dimensional quadratic forms and Rost's theorem on 14-dimensional quadratic forms. This material on quadratic forms and invariants of $\mathrm{Spin}_n$ is based on unpublished work of Markus Rost. An appendix by Detlev Hoffmann proves a generalization of the Common Slot Theorem for 2-Pfister quadratic forms.A. Examples of anisotropic groups of type E7 We use cohomological invariants to give examples of algebraic groups of type E7 that ... The Rost invariant rE7 recalled in Example 1.2.4 maps rE7: H1(a, E 7) a H3(a, Z/12Z(2)), see [Mer, pp. ... We consult the list of the possible Tits indexes of groups of type E7 from [Tits 66 , p.

 Title : Cohomological Invariants Author : Skip Garibaldi Publisher : American Mathematical Soc. - 2009-06-05