The positive scalar curvature cobordism category
نویسندگان
چکیده
We prove that many spaces of positive scalar curvature (psc) metrics have the homotopy type infinite loop spaces. Our result in particular applies to path component round metric inside R+(Sd) if d≥6. To achieve goal, we study cobordism category manifolds with curvature. Under suitable connectivity conditions, can identify fiber forgetful map from psc ordinary a delooping metrics. This uses version Quillen’s Theorem B and instances Gromov–Lawson surgery theorem. extend some arguments by Galatius second-named author setting pass between different conditions. Segal’s theory Γ-spaces is then used construct claimed space structures. The viewpoint also illuminates action diffeomorphism groups on show under mild hypotheses manifold, group automorphisms factors through Madsen–Tillmann spectrum. implies strong rigidity theorem for when manifold has trivial rational Pontryagin classes. A delooped Atiyah–Singer index proved first-named is, moreover, secondary invariant real K-theory an map. These ideas give new proof main our previous work Botvinnik.
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ژورنال
عنوان ژورنال: Duke Mathematical Journal
سال: 2022
ISSN: ['1547-7398', '0012-7094']
DOI: https://doi.org/10.1215/00127094-2022-0023