The components of a variety of matrices with square zero and submaximal rank

A longstanding conjecture in algebraic topology describes the free rank of symmetry of a product of spheres. This conjecture states that if the elementary abelian group pr acts freely on Sn1 Sns , then r s. A more ambitious generalization of this conjecture states that if pr acts freely on a manifold M, then ∑i dimp HiMp 2r. For a survey of conjectures of this type, and partial results, the reader might consult Section 2 of [1].