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Complementing and extending the Inventiones work of Benson, Grodal, Henke[Group cohomology and control of p-fusion, Invent. Math. 197 (2014), 491--507]we give criteria for a space to have cohomology (strongly) F-isomorphic in thesense of Quillen to the stable elements. We extend results about groups modelsto fusion systems over discrete $p$-toral groups and profinite groups andprovide various applications to the Kan-Thurston Theorem.

Assigning a classifying space to a fusion system up to F-isomorphism