Cohomology and Extensions
Formal Lie groupoids
The Lie group called is the largest-dimensional one of the five exceptional Lie groups.
The first nontrivial homotopy group of the topological space underlying is
as for any compact Lie group. Then the next nontrivial homotopy group is
This means that all the way up to the 15 coskeleton the group looks, homotopy theoretically like the Eilenberg-MacLane space .
By the above discussion of homotopy groups, it follows (by Chern-Weil theory) that the first invariant polynomials on the Lie algebra are the quadratic Killing form and then next an octic polynomial. That is described in (Cederwall-Palmkvist).
The gorup plays a role in some exceptional differential geometry/differential cohomology. See for instance
exceptional generalized geometry, supergravity C-field, Hořava-Witten theory, heterotic string theory
E6, E7, E8, E9, E10, E11,
An introductory survey with an eye towards the relation to the octonions is given in section 4.6 of
The lower homotopy groups of are a classical result due to
- Raoul Bott and H. Samelson, Application of the theory of Morse to symmetric spaces , Amer. J. Math., 80 (1958), 964-1029.
The higher homotopy groups are discussed in
- Hideyuki Kachi, Homotopy groups of compact Lie groups , and Nagoya Math. J. Volume 32 (1968), 109-139. (project EUCLID)
The octic invariant polynomial is discussed in
Revised on April 15, 2013 07:37:12
by David Roberts