# Cohomological Invariants: Exceptional Groups and Spin Groups

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*Skip Garibaldi*

with an appendix by Detlev Hoffmann

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unpublished work of Markus Rost.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.

#### Table of Contents

# Table of Contents

## Cohomological Invariants: Exceptional Groups and Spin Groups

- Contents v6 free
- List of Tables ix10 free
- Preface xi12 free
- Part I. Invariants, especially modulo an odd prime 114 free
- 1. Definitions and notations 215
- 2. Invariants of μ[sub(n)] 518
- 3. Quasi-Galois extensions and invariants of Z/pZ 720
- 4. An example: the mod p Bockstein map 1023
- 5. Restricting invariants 1225
- 6. Mod p invariants of PGL[sub(p)] 1427
- 7. Extending invariants 1730
- 8. Mod 3 invariants of Albert algebras 1932

- Part II. Surjectivities and invariants of E[sub(6)], E[sub(7)], and E[sub(8)] 2336
- 9. Surjectivities: internal Chevalley modules 2437
- 10. New invariants from homogeneous forms 2942
- 11. Mod 3 invariants of simply connected E[sub(6)] 3144
- 12. Surjectivities: the highest root 3346
- 13. Mod 3 invariants of E[sub(7)] 3851
- 14. Construction of groups of type E[sub(8)] 3952
- 15. Mod 5 invariants of E[sub(8)] 4457

- Part III. Spin groups 4760
- 16. Introduction to Part III 4861
- 17. Surjectivities: Spin[sub(n)] for 7 ≤ n ≤ 12 4861
- 18. Invariants of Spin[sub(n)] for 7 ≤ n ≤ 10 5366
- 19. Divided squares in the Grothendieck-Witt ring 5669
- 20. Invariants of Spin[sub(11)] and Spin[sub(12)] 5871
- 21. Surjectivities: Spin[sub(14)] 6174
- 22. Invariants of Spin[sub(14)] 6578
- 23. Partial summary of results 6679

- Appendices 6982
- Bibliography 7790
- Index 8194 free