Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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

Author: Houhong Fan
Journal: Trans. Amer. Math. Soc. 348 (1996), 2947-2982
MSC (1991): Primary 57R25, 57M99; Secondary 57R80, 58F25, 58A12
MathSciNet review: 1357879
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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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Additional Information

Houhong Fan
Affiliation: Department of Mathematics, Yale University, New Haven, Connecticut 06520

Received by editor(s): March 23, 1995
Received by editor(s) in revised form: November 6, 1995
Article copyright: © Copyright 1996 American Mathematical Society