On equivariant algebraic suspension

Equivariant versions of the Suspension Theorem [L₁] for algebraic cycles on projective varieties are proved. Let G be a finite group, V a complex G-module, and X ⊂ ℙ_ℂ(V) an invariant subvariety. Consider the algebraic join Σ^V0 X = X#ℙ_ℂ(V₀) of X with the regular representation V₀ = ℂ^G of G. The main result asserts that algebraic suspension induces a G-homotopy equivalence Z^s (X) → Z^s (Σ^V0X) 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-fixed point sets in Σ^V0X for g ≠ 1. This leads to a Stability Theorem for equivariant algebraic suspension. The result enables the determination of coefficients in certain 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(ℍⁿ) under the antiholomorphic involution corresponding to scalar multiplication by the quaternion j. From this the homotopy type of spaces of quaternionic cycles is determined.