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Nontame Morse–Smale flows and odd Chern–Weil theory

Part of: Manifolds Global differential geometry General theory of differentiable manifolds

Published online by Cambridge University Press:  06 July 2021

Daniel Cibotaru  and
Wanderley Pereira
Daniel Cibotaru*
Affiliation:
Departamento de Matematicash ort Universidade Federal do Ceará, Fortaleza, CE, Brazil
Wanderley Pereira
Affiliation:
Universidade Estadual do Ceará, Limoeiro do Norte, CE, Brazil e-mail: wanderley.pereira@uece.br
*
e-mail: daniel@mat.ufc.br
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Abstract

Using a certain well-posed ODE problem introduced by Shilnikov in the sixties, Minervini proved the currential “fundamental Morse equation” of Harvey–Lawson but without the restrictive tameness condition for Morse gradient flows. Here, we construct local resolutions for the flow of a section of a fiber bundle endowed with a vertical vector field which is of Morse gradient type in every fiber in order to remove the tameness hypothesis from the currential homotopy formula proved by the first author. We apply this to produce currential deformations of odd degree closed forms naturally associated to any hermitian vector bundle endowed with a unitary endomorphism and metric compatible connection. A transgression formula involving smooth forms on a classifying space for odd K-theory is also given.

Keywords

Morse equationgradient flowsodd Chern-Weiltransgressioncurrential identityflow resolution

MSC classification

Primary: 58A25: Currents
Secondary: 49Q15: Geometric measure and integration theory, integral and normal currents 53C05: Connections, general theory
Type
Article
Information
Canadian Journal of Mathematics , Volume 74 , Issue 6 , December 2022 , pp. 1579 - 1624
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Copyright
© Canadian Mathematical Society 2021

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Footnotes

The first author was partially supported by the CNPq Projeto Universal 427191/2016-5

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