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Stratification and duality for homotopical groups. (English) Zbl 1426.55017
Let $$G$$ be a finite group. Some of the most celebrated results in group cohomology are the $$F$$-isomorphism theorem of Quillen and the study of the spectrum of the cohomology ring of $$G$$ with coefficients in a field $$k$$ whose characteristic divides the order of $$G$$. There have been generalizations of these results to other instances like $$p$$-compact groups and $$p$$-local finite groups. The main objective of this paper is to generalize these (and many related) results to $$p$$-local compact groups. Much of the work is devoted to establishing the technical tools to achieve these results. We provide some examples. A $$p$$-local compact group $$\mathcal{G}=(S,\mathcal{F})$$ satisfies Choinard’s theorem if induction and coinduction along the morphism induced by restriction $C^{*}(B\mathcal{G},k)\to \prod_{\mathcal{E}(\mathcal{G})} C^{*}(BE,k),$ is conservative, where $$\mathcal{E}(\mathcal{G})$$ denotes a set of representatives of $$\mathcal{F}$$-conjugacy classes of elementary abelian subgroups of $$S$$ and $$C^{*}(B(-),k)$$ is the spectrum of $$k$$-valued cochains on $$B(-)$$. The authors prove that connected $$p$$-local groups satisfy Choinard’s theorem. They also prove the generalization of Quillen’s $$F$$-isomorphism theorem to these groups:
Theorem. Let $$\mathcal{G}=(S,\mathcal{F})$$ be a $$p$$-local compact group, the the restriction to elementary abelian subgroups of $$S$$ induces an $$F$$-isomorphism $H^{*}(B\mathcal{G},\mathbf{F}_p)\to \lim_{\mathcal{F}^{e}} H^{*}(BE,\mathbb{F}_p),$ where $$\mathbb{F}^{e}$$ denotes the full subcategory of $$\mathbb{F}$$ consisting elementary abelian $$p$$-groups.
They also prove that $$p$$-local compact groups satisfy Quillen’s stratification.

##### MSC:
 55R35 Classifying spaces of groups and $$H$$-spaces in algebraic topology 20J05 Homological methods in group theory 13D45 Local cohomology and commutative rings 55P42 Stable homotopy theory, spectra
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