Torsion in equivariant cohomology and Cohen-Macaulay G-actions

Abstract

Abstract. We show that the well-known fact that the equivariant cohomology of a torus action is a torsion-free module if and only if the map induced by the inclusion of the fixed point set is injective generalises to actions of arbitrary compact connected Lie groups if one replaces the fixed point set by the set of points with maximal isotropy rank. This is true essentially because the action on this set is always equivariantly formal.In case this set is empty we show that the induced action on the set of points with highest occuring isotropy rank is Cohen-Macaulay. It turns out that just as equivariant formality of an action is equivalent to equivariant formality of the action of a maximal torus, the same holds true for equivariant injectivity and the Cohen-Macaulay property. In addition, we find a topological criterion for equivariant injectivity in terms of orbit spaces.

Equivariant de Rham cohomology: theory and applications

“…There are various other ways to obtain this theorem, without using the Bruhat decomposition. Given a homogeneous space G/H of equal rank, all isotropy groups of the Gaction on H have the same rank as that of G. For such actions equivariant formality is automatic, see [43,Proposition 3.7]. Then, Proposition 10.1 and Theorem 7.3 imply the description of the cohomology ring.…”

Syzygies in Equivariant Cohomology for Non-abelian Lie Groups

“…Like q, the map f 3 is a homeomorphism. Again by [22,Lemma 3.2], the fibre K N K (T ) of the bundle map f 2 is acyclic, so that f 2 is a quasi-isomorphism. We finally know from Lemma 5.4 that f 1 is a quasi-isomorphism, too.…”

“…It turns out that essentially all results carry over to this more general setting. We achieve this by combining the techniques of Allday-Franz-Puppe with those of Goertsches-Rollenske [22], whose study of Cohen-Macaulay actions gave a first hint at the possibility of such an extension. Let us describe our results in more detail.…”

Non-abelian GKM theory

“…was proven in [7,Proposition 4.8]. The exactness at H * G (M max ) can be proven with the same method as the exactness of the Atiyah-Bredon sequence for torus actions [2, Main Lemma], see also [4].…”

“…However this is unclear, as to copy the proof given there one needs to know that the modules H * G (M i , M i+1 ) are Cohen-Macaulay of dimension i. This is so far only proven for i = r: [7,Corollary 4.5] shows that H * G (M r , M r+1 ) = H * G (M max ) is a free module over H * (BG).…”