## Journal of the Mathematical Society of Japan

### The finite group action and the equivariant determinant of elliptic operators II

Kenji TSUBOI

#### Abstract

Let $M$ be an almost complex manifold and $g$ a periodic automorphism of $M$ of order $p$. Then the rotation angles of $g$ around fixed points of $g$ are naturally defined by the almost complex structure of $M$. In this paper, under the assumption that the fixed points of $g^k$ $(1\leq k\leq p-1)$ are isolated, a calculation formula is provided for the homomorphism $I_D: {\Bbb Z}_p \to {\Bbb R}/{\Bbb Z}$ defined in [8]. The formula gives a new method to study the periodic automorphisms of almost complex manifolds. As examples of the application of the formula, we show the nonexistence of the ${\Bbb Z}_p$-action of specific isotropy orders and examine whether specific rotation angles exist or not.

#### Article information

Source
J. Math. Soc. Japan, Volume 65, Number 3 (2013), 797-827.

Dates
First available in Project Euclid: 23 July 2013

https://projecteuclid.org/euclid.jmsj/1374586627

Digital Object Identifier
doi:10.2969/jmsj/06530797

Mathematical Reviews number (MathSciNet)
MR2114722

Zentralblatt MATH identifier
1276.58011

#### Citation

TSUBOI, Kenji. The finite group action and the equivariant determinant of elliptic operators II. J. Math. Soc. Japan 65 (2013), no. 3, 797--827. doi:10.2969/jmsj/06530797. https://projecteuclid.org/euclid.jmsj/1374586627

#### References

• M. F. Atiyah and G. B. Segal, The index of elliptic operators. II, Ann. of Math. (2), 87 (1968), 531–545.
• M. F. Atiyah and I. M. Singer, The index of elliptic operators. III, Ann. of Math. (2), 87 (1968), 546–604.
• E. Bujalance, F. J. Cirre, J. M. Gamboa and G. Gromadzki, On compact Riemann surfaces with dihedral groups of automorphisms, Math. Proc. Cambridge Philos. Soc., 134 (2003), 465–477.
• H. Glover and G. Mislin, Torsion in the mapping class group and its cohomology, J. Pure Appl. Algebra, 44 (1987), 177–189.
• W. J. Harvey, Cyclic groups of automorphisms of a compact Riemann surface, Quart. J. Math. Oxford Ser. (2), 17 (1966), 86–97.
• F. Hirzebruch, Topological Methods in Algebraic Geometry, 3rd ed., Grundlehren Math. Wiss., 131, Springer-Verlag, Berlin, Heidelberg, New York, 1966.
• S. P. Kerckhoff, The Nielsen realization problem, Ann. of Math. (2), 117 (1983), 235–265.
• K. Tsuboi, The finite group action and the equivariant determinant of elliptic operators, J. Math. Soc. Japan, 57 (2005), 95–113.
• A. Van de Ven, On the Chern numbers of certain complex and almost complex manifolds, Proc. Natl. Acad. Sci. U.S.A., 55 (1966), 1624–1627.
• D. B. Zagier, Equivariant Pontrjagin Classes and Applications to Orbit Spaces, Lecture Notes in Math., 290, Springer-Verlag, Berlin, Heidelberg, New York, 1972.