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Vector bundles over classifying spaces of $$p$$-local finite groups and Benson-Carlson duality. (English) Zbl 1444.55007
This paper describes the Grothendieck group of complex vector bundles over the classifying space of a $$p$$-local finite group $$(S,\mathcal{F},\mathcal{L})$$ in terms of representation rings of the subgroups of $$S$$. Here $$S$$ is a finite $$p$$-group, $$\mathcal{F}$$ is a saturated fusion system over $$S$$, and $$\mathcal{L}$$ is a centric linking system associated to $$\mathcal{F}$$. The classifying space of $$(S,\mathcal{F},\mathcal{L})$$ is $$\vert\mathcal{L} \vert _{p}^{\wedge}$$ and the main result of the paper shows the existence of an isomorphism between $${\mathbb K}(\vert\mathcal{L} \vert _{p}^{\wedge})$$ and an inverse limit of the representation rings of the subgroups of $$S$$. The authors show also that the cohomology of the classifying space $$\vert\mathcal{L} \vert _{p}^{\wedge}$$ is isomorphic to the inverse limit of the cohomology of the subgroups of $$S$$, for any generalized cohomology theory. The authors also show that there exists a homotopy monomorphism $$\vert\mathcal{L} \vert _{p}^{\wedge}\rightarrow\mathrm{BSU}(n)_{p}^{\wedge}$$ for some $$n$$ whose homotopy fibre is an $${\mathbb F}_{p}$$-finite space with Poincaré duality.

##### MSC:
 55R35 Classifying spaces of groups and $$H$$-spaces in algebraic topology 20C20 Modular representations and characters 20D20 Sylow subgroups, Sylow properties, $$\pi$$-groups, $$\pi$$-structure
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