A direct geometric proof of the Lefschetz fixed point formulas

Lafferty, John D.; Yu, Yan Lin; Weiping, Zhang

doi:10.1090/S0002-9947-1992-1022168-9

A direct geometric proof of the Lefschetz fixed point formulas
HTML articles powered by AMS MathViewer

by John D. Lafferty, Yan Lin Yu and Zhang Weiping PDF

Trans. Amer. Math. Soc. 329 (1992), 571-583 Request permission

Abstract:

In this paper we prove the Lefschetz fixed point formulas of Atiyah, Singer, Segal, and Bott for isometries by using the direct geometric method initiated by Patodi.

References

M. F. Atiyah and R. Bott, A Lefschetz fixed point formula for elliptic complexes. I, Ann. of Math. (2) 86 (1967), 374–407. MR 212836, DOI 10.2307/1970694
M. F. Atiyah and R. Bott, A Lefschetz fixed point formula for elliptic complexes. II. Applications, Ann. of Math. (2) 88 (1968), 451–491. MR 232406, DOI 10.2307/1970721
Nicole Berline and Michèle Vergne, A computation of the equivariant index of the Dirac operator, Bull. Soc. Math. France 113 (1985), no. 3, 305–345 (English, with French summary). MR 834043
Jean-Michel Bismut, The Atiyah-Singer theorems: a probabilistic approach. I. The index theorem, J. Funct. Anal. 57 (1984), no. 1, 56–99. MR 744920, DOI 10.1016/0022-1236(84)90101-0
Ezra Getzler, A short proof of the local Atiyah-Singer index theorem, Topology 25 (1986), no. 1, 111–117. MR 836727, DOI 10.1016/0040-9383(86)90008-X
Mark W. Goodman, Proof of character-valued index theorems, Comm. Math. Phys. 107 (1986), no. 3, 391–409. MR 866197
V. K. Patodi, An analytic proof of Riemann-Roch-Hirzebruch theorem for Kaehler manifolds, J. Differential Geometry 5 (1971), 251–283. MR 290318
V. K. Patodi, Curvature and the eigenforms of the Laplace operator, J. Differential Geometry 5 (1971), 233–249. MR 292114
Yan Lin Yu, Local index theorem for Dirac operator, Acta Math. Sinica (N.S.) 3 (1987), no. 2, 152–169. MR 913775, DOI 10.1007/BF02565028
Wei Ping Zhang, Local Atiyah-Singer index theorem for families of Dirac operators, Differential geometry and topology (Tianjin, 1986–87) Lecture Notes in Math., vol. 1369, Springer, Berlin, 1989, pp. 351–366. MR 1001201, DOI 10.1007/BFb0087548