Abstract
We investigate the topological nilpotence degree, in the sense of Henn–Lannes–Schwartz, of a connected Noetherian unstable algebra R. When R is the mod p cohomology ring of a compact Lie group, Kuhn showed how this invariant is controlled by centralizers of elementary abelian p-subgroups. By replacing centralizers of elementary abelian p-subgroups with components of Lannes’ T-functor, and utilizing the techniques of unstable algebras over the Steenrod algebra, we are able to generalize Kuhn’s result to a large class of connected Noetherian unstable algebras. We show how this generalizes Kuhn’s result to more general classes of groups, such as groups of finite virtual cohomological dimension, profinite groups, and Kac–Moody groups. In fact, our results apply much more generally, for example, we establish results for p-local compact groups in the sense of Broto–Levi–Oliver, for connected H-spaces with Noetherian mod p cohomology, and for the Borel equivariant cohomology of a compact Lie group acting on a manifold. Along the way we establish several results of independent interest. For example, we formulate and prove a version of Carlson’s depth conjecture in the case of a Noetherian unstable algebra of minimal depth.
Highlights
1.1 Motivation and overviewWhen G is a compact Lie group, or even just a finite group, the mod p cohomology ring HG∗ :=H ∗(BG; Fp) can be extremely complicated
[34] Henn, Lannes, and Schwartz gave a rough upper bound for d0(HG∗ (X )), the mod p Borel-equivariant cohomology of a compact Lie group G acting on a manifold
The proof of Theorem B follows the same strategy as Kuhn; we show in Proposition 5.17 that for any connected Noetherian unstable algebra R with center (C, g) we have d0 ( R )
Summary
When G is a compact Lie group, or even just a finite group, the mod p cohomology ring HG∗ :=H ∗(BG; Fp) can be extremely complicated. Quillen introduced the category AG of elementary abelian psubgroups of G, with morphisms those group homomorphisms induced by conjugation in G He proved that the restriction maps induced a morphism q1 : HG∗. In [34] Henn, Lannes, and Schwartz gave a rough upper bound for d0(HG∗ (X )), the mod p Borel-equivariant cohomology of a compact Lie group G acting on a manifold. The product is taken over those elementary abelian p-subgroups E of G for which C(G) is strictly contained in E, and the map is the map induced by the inclusions CG (E) ≤ G He shows that d0(HG∗ ) = max{d0(CEss(CG (E))) | E < G}.
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