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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Self-duality operators on
odd dimensional manifolds


Author: Houhong Fan
Journal: Trans. Amer. Math. Soc. 350 (1998), 3673-3706
MSC (1991): Primary 57R25, 58G10; Secondary 57R80, 57M99, 58F25
MathSciNet review: 1422603
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Abstract: In this paper we construct a new elliptic operator associated to any nowhere zero vector field on an odd-dimensional manifold and study its index theory. It turns out this operator has several geometric applications to conformal vector fields, self-dual vector fields, locally free $S^{1}$-actions and transversal hypersurfaces of these vector fields in an odd-dimensional manifold. In particular, we reveal a non-stable phenomena about the existence of conformal vector fields and self-dual vector fields in odd dimensions above 3. This is in sharp contrast to the stable phenomena about the existence of nowhere zero vector fields in odd dimensions. Besides these applications, the index formula of this new operator also gives the formulas for the dimensions of self-duality cohomology groups and for the virtual dimensions of the moduli spaces of anti-self-dual connections on 5-cobordisms, which are introduced in author's previous papers.


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Additional Information

Houhong Fan
Affiliation: Department of Mathematics, Yale University, New Haven, Connecticut 06520
Email: hhfan@math.yale.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-98-01954-0
PII: S 0002-9947(98)01954-0
Received by editor(s): August 18, 1995
Received by editor(s) in revised form: October 6, 1996
Article copyright: © Copyright 1998 American Mathematical Society