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Cohomological Invariants: Exceptional Groups and Spin Groups
Skip Garibaldi, Emory University, Atlanta, GA
with an appendix by Detlev Hoffmann
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Memoirs of the American Mathematical Society
2009; 81 pp; softcover
Volume: 200
ISBN-10: 0-8218-4404-0
ISBN-13: 978-0-8218-4404-5
List Price: US$65 Individual Members: US$39
Institutional Members: US\$52
Order Code: MEMO/200/937

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 This material on quadratic forms and invariants of $$\mathrm{Spin}_n$$ is based on unpublished work of Markus Rost.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.

• Part II. Surjectivities and invariants of $$E_6, E_7$$, and $$E_8$$