Half De Rham complexes and line fields on odd-dimensional manifolds

Fan, Houhong

doi:10.1090/S0002-9947-96-01661-3

Half De Rham complexes and line fields on odd-dimensional manifolds
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by Houhong Fan

Trans. Amer. Math. Soc. 348 (1996), 2947-2982

DOI: https://doi.org/10.1090/S0002-9947-96-01661-3

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Abstract:

In this paper we introduce a new elliptic complex on an odd-dimensional manifold with a self-dual line field. The notion of a self-dual line field is a generalization of the notion of a conformal line field. Ellipticity, Fredholm properties and Hodge decompositions of these new complexes are proved both in the case of a closed manifold and in the case of a manifold with boundary. The cohomology groups of these elliptic complexes are computed in some cases. In addition, in this paper, we generalize the notion of an anti-self-dual connection on a smooth 4-manifold to a 3-manifold with a line field and a smooth 5-manifold with a line field. The above new elliptic complexes can be twisted by anti-self-dual connections in dimensions 3 and 5, but only by flat connections in dimensions above 5. This reveals a special feature of dimensions 3 and 5.

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