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Homotopy invariance of cohomology and signature of a Riemannian foliation

We prove that any smooth foliation that admits a Riemannian foliation structure has a well-defined basic signature, and this geometrically defined invariant is actually a foliated homotopy invariant. We also show that foliated homotopic maps between Riemannian foliations induce isomorphic maps on ba...

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Bibliographic Details
Published in:Mathematische Zeitschrift 2019-10, Vol.293 (1-2), p.579-595
Main Authors: Habib, Georges, Richardson, Ken
Format: Article
Language:English
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Summary:We prove that any smooth foliation that admits a Riemannian foliation structure has a well-defined basic signature, and this geometrically defined invariant is actually a foliated homotopy invariant. We also show that foliated homotopic maps between Riemannian foliations induce isomorphic maps on basic Lichnerowicz cohomology, and that the Álvarez class of a Riemannian foliation is invariant under foliated homotopy equivalence.
ISSN:0025-5874
1432-1823
DOI:10.1007/s00209-018-2195-x