Transversality and Lefschetz numbers for foliation maps
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Publication:948485
DOI10.1007/S11784-008-0071-8zbMath1151.57034arXiv0801.4628OpenAlexW2042096361MaRDI QIDQ948485
Jesús A. Álvarez López, Yurii A. Kordyukov
Publication date: 16 October 2008
Published in: Journal of Fixed Point Theory and Applications (Search for Journal in Brave)
Abstract: Let $F$ be a smooth foliation on a closed Riemannian manifold $M$, and let $Lambda$ be a transverse invariant measure of $F$. Suppose that $Lambda$ is absolutely continuous with respect to the Lebesgue measure on smooth transversals. Then a topological definition of the $Lambda$-Lefschetz number of any leaf preserving diffeomorphism $(M,F) o(M,F)$ is given. For this purpose, standard results about smooth approximation and transversality are extended to the case of foliation maps. It is asked whether this topological $Lambda$-Lefschetz number is equal to the analytic $Lambda$-Lefschetz number defined by Heitsch and Lazarov which would be a version of the Lefschetz trace formula. Heitsch and Lazarov have shown such a trace formula when the fixed point set is transverse to $F$.
Full work available at URL: https://arxiv.org/abs/0801.4628
foliationtransversalityLefschetz trace formula\(\Lambda\)-Lefschetz numberfoliation mapintegrable homotopy
Index theory and related fixed-point theorems on manifolds (58J20) Foliations in differential topology; geometric theory (57R30)
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Transversal multiplicities of the Lyashko-Looijenga map ⋮ The higher fixed point theorem for foliations. I: Holonomy invariant currents
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