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Cohomology of Horizontal Forms

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Abstract

The complex of s-horizontal forms of a smooth foliation F on a manifold M is proved to be exact for every s = 1, . . . , n = codim F, and the cohomology groups of the complex of its global sections, are introduced. They are then compared with other cohomology groups associated to a foliation, previously introduced. An explicit formula for an s-horizontal primitive of an s-horizontal closed form, is given. The problem of representing a de Rham cohomology class by means of a horizontal closed form is analysed. Applications of these cohomology groups are included and several specific examples of explicit computation of such groups—even for non-commutative structure groups—are also presented.

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Authors and Affiliations

  1. Instituto de Física Aplicada, CSIC, C/ Serrano 144, 28006, Madrid, Spain

    J. Muñoz Masqué

  2. Departamento de Geometría y Topología, Universidad Complutense de Madrid, Ciudad Universitaria s/n, 28040, Madrid, Spain

    L. M. Pozo Coronado

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  1. J. Muñoz Masqué
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  2. L. M. Pozo Coronado
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Correspondence to L. M. Pozo Coronado.

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Supported by MICINN (Spain), under grant # MTM2008–01386.

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Muñoz Masqué, J., Pozo Coronado, L.M. Cohomology of Horizontal Forms. Milan J. Math. 80, 169–202 (2012). https://doi.org/10.1007/s00032-012-0173-z

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