On Equivariant Algebraic Suspension

Equivariant versions of the Suspension Theorem L 1 ] for algebraic cycles on projective varieties are proved. Let G be a nite group, V a projective G-module, and X P C (V) an invariant subvariety. Consider the algebraic join = V 0 X = X#P C (V 0) of X with the 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 ((= V 0 X) of topological groups of algebraic cycles of codimension-s for all s dim X ? e(X) where e(X) is the maximal dimension of g-xed point sets in = V 0 X for g 6 = 1. This leads to a Stability Theorem for equivariant algebraic suspension. The result enables the determination of coeecients in certain equivariant cohomology theories based on algebraic cycles, and it enables the deenition 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 multiplication by the quaternion j. From this the homotopy type of spaces of quaternionic cycles is determined.