ON EQUIVARIANT ALGEBRAIC SUSPENSION

Equivariant versions of the Suspension Theorem [L-1] for algebraic cyc les on projective varieties are proved. Let G be a finite group, V a c omplex G-module, and X subset of P-C(V) an invariant subvariety. Consi der the algebraic join <(Sigma)over bar>X-V0 = X#P-C(V-0) of X with th e regular representation V-0 = C-G Of G. The main result asserts that algebraic suspension induces a G-homotopy equivalence Z(s)(X) --> Z(s) (Sigma(V0)X) of topological groups of algebraic cycles of codimension- s for all s less than or equal to dim X - e(X) where e(X) is the maxim al dimension of g-fixed point sets in <(Sigma)over bar>X-V0 for g not equal 1. This leads to a Stability Theorem for equivariant algebraic s uspension. The result enables the determination of coefficients in cer tain equivariant cohomology theories based on algebraic cycles, and it enables the definition of cohomology operations in such theories. The methods also yield a Quaternionic Suspension Theorem for cycles in P- C(H-n) under the antiholomorphic involution corresponding to scalar mu ltiplication by the quaternion j. From this the homotopy type of space s of quaternionic cycles is determined.