The stratification of compact connected Lie group actions by subtori

Abstract

A stratification of the spectrum of the mod p equivariant cohomology ring of a compact Lie group action in terms of elementary p-subgroups has been obtained by Quillen.A corresponding result for compact connected Lie group actions in terms of subtori is proven in this paper by different means.In addition some localization and primary decomposition theorems are obtained.The paper closes with an application to uniform torus actions. Introduction. In [11]Quillen proves a stratification theorem for compact Lie group actions by elementary abelian p -subgroups in equivariant cohomology with coefficients in Z/pZ.In the first section of this paper we prove the stratification theorem for compact connected Lie group actions by subtori in equivariant cohomology with rational coefficients.We do not attempt to follow Quillen's method of proof, however.There are three reasons for this.First, the existence in Quillen's situation of a "universal invertible" (the element e A of Theorem 4.2 of [10]) has no natural analogue in our situation.Secondly, Quillen's vital Main Theorem (Theorem 6.2 of [10]) appears to require certain restrictions on the orbit structure which can be avoided.(See Lemma 1.5, below).Thirdly, the results and techniques of Hsiang, Chang and Skjelbred ([7], [6] and [13]) give rise to a proof which is more direct and less sophisticated than the proof Quillen gives for his theorem.In the second section of this paper, we prove a localization theorem for H τ (X)-module structures and deduce analogues of the results of [6], using H τ (X)-module structures instead of iτΓ(J5 τ )-module structures.The advantage of this approach is that it distinguishes between components of the fixed point set of a subtorus.In the third section, by way of an application, the useful concept of a uniform torus action is defined, and a simple algebraic characterization of uniformity is given.Throughout this paper the cohomology and equivariant cohomology theory used will be sheaf theoretic, or equivalently, Cech.Rational coefficients will be used throughout, and these will be suppressed from the notation.The form of the "going up" theorem of Cohen and Seidenberg, which is used in the first section, is that which may be found in Serre ([12]).We wish to thank Professor W.-Y. Hsiang, T. Chang and T. Skjelbred for many useful discussions and suggestions.