On the geometric weight system of topological actions on cohomology quaternionic projective spaces

Abstract

In this paper, we take those cohomology manifolds X of rational cohomology type of quaternionic projective spaces as the testing spaces for the study of cohomology theory of topological transformation groups. We follow the general viewpoint set out in [-3, 4] of seeking for the socalled topological Schur lemma in the form of splitting theorem in equivariant cohomology for torus group actions on cohomology quaternionic projective spaces. The main result of this paper is precisely such a structural theorem, Theorem(2.1), which proves that topological actions of torus groups (of rank >__ 2) have exactly the same cohomological behavior as that of suitable linear models. Namely, for a cohomology manifold X of QP-type with a given effective action of a torus group G (rank _>2, or 3 if all fixed points are isolated), there exists a unique lifting ( of the generator (o of H * ( X ) into H G (X) such that