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
HTML articles powered by AMS MathViewer

by Houhong Fan PDF

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

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.

References

Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
N. Aronszajn, A. Krzywicki, and J. Szarski, A unique continuation theorem for exterior differential forms on Riemannian manifolds, Ark. Mat. 4 (1962), 417–453 (1962). MR 140031, DOI 10.1007/BF02591624
Richard S. Palais, Seminar on the Atiyah-Singer index theorem, Annals of Mathematics Studies, No. 57, Princeton University Press, Princeton, N.J., 1965. With contributions by M. F. Atiyah, A. Borel, E. E. Floyd, R. T. Seeley, W. Shih and R. Solovay. MR 0198494, DOI 10.1515/9781400882045
M. F. Atiyah and I. M. Singer, The index of elliptic operators. III, Ann. of Math. (2) 87 (1968), 546–604. MR 236952, DOI 10.2307/1970717
Lawrence Conlon, Transversally parallelizable foliations of codimension two, Trans. Amer. Math. Soc. 194 (1974), 79–102, erratum, ibid. 207 (1975), 406. MR 370617, DOI 10.1090/S0002-9947-1974-0370617-0
S. K. Donaldson and P. B. Kronheimer, The geometry of four-manifolds, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1990. Oxford Science Publications. MR 1079726
H.H. Fan, Self-dual vector fields on odd dimensional manifolds. Preprint, Yale, 1995.
H.H. Fan, Gauge fields and line fields on five-manifolds. Preprint, Yale, 1995.
H.H. Fan, Seiberg-Witten-type equation on five-manifolds. Preprint, Yale, 1995.
J.F. Glazebrook, F.W. Kamber, H. Pedersen and A. Swann, Foliation reduction and self-duality, Geometric study of foliations, World Scientific Press, Singapore, 1994.
Pierre Molino, Riemannian foliations, Progress in Mathematics, vol. 73, Birkhäuser Boston, Inc., Boston, MA, 1988. Translated from the French by Grant Cairns; With appendices by Cairns, Y. Carrière, É. Ghys, E. Salem and V. Sergiescu. MR 932463, DOI 10.1007/978-1-4684-8670-4
Seiki Nishikawa and Hajime Sato, On characteristic classes of Riemannian, conformal and projective foliations, J. Math. Soc. Japan 28 (1976), no. 2, 223–241. MR 400255, DOI 10.2969/jmsj/02820223
D. B. Ray and I. M. Singer, $R$-torsion and the Laplacian on Riemannian manifolds, Advances in Math. 7 (1971), 145–210. MR 295381, DOI 10.1016/0001-8708(71)90045-4
Izu Vaisman, Conformal foliations, Kodai Math. J. 2 (1979), no. 1, 26–37. MR 531785