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Rings of Generalized and Stable Invariants and Classifying Spaces of Compact Lie Groups

by Neumann, F., Neusel, M.D., Smith, L..

Series: 1996-14, Preprints

13A50 Actions of groups on commutative rings; invariant theory
55R40 Homology of classifying spaces, characteristic classes
57T10 Homology and cohomology of Lie groups

G be a compact connected Lie group with maximal torus T ,!
G and Weyl group WG . There is the fibration
G j
\Gamma! G=T k
\Gamma! BT
and we show that the Eilenberg-Moore spectral sequence mod p, for
an odd prime p, of this fibration collapses at the term E 2 . As an
important step in the proof we compute the kernel of the induced
ker f(k \Lambda ) : H \Lambda (BT ; F p ) \Gamma! H \Lambda (G=T; F p )g
and identify it with the ideal J1 (WG ) ae H \Lambda (BT ; F p ) of stable in-
variants of the Weyl group. We apply our results to determine by
means of the action of WG on H \Lambda (BT ; F p ) the odd primes p for
which H \Lambda (BG; Z) has p-torsion.