Equivariant method for periodic maps

Author:
Wu Hsiung Huang

Journal:
Trans. Amer. Math. Soc. **189** (1974), 175-183

MSC:
Primary 57D70

DOI:
https://doi.org/10.1090/S0002-9947-1974-0334247-9

MathSciNet review:
0334247

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Abstract: The notion of coherency with submanifolds for a Morse function on a manifold is introduced and discussed in a general way. A Morse inequality for a given periodic transformation which compares the invariants called *q*th Euler numbers on fixed point set and the invariants called *q*th Lefschetz numbers of the transformations is thus obtained. This gives a fixed point theorem in terms of *q*th Lefschetz number for arbitrary *q*.

**[1]**Shiing-shen Chern,*A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian manifolds*, Ann. of Math. (2)**45**(1944), 747–752. MR**0011027**, https://doi.org/10.2307/1969302**[2]**Shoshichi Kobayashi,*Fixed points of isometries*, Nagoya Math. J.**13**(1958), 63–68. MR**0103508****[3]**J. Milnor,*Morse theory*, Based on lecture notes by M. Spivak and R. Wells. Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, N.J., 1963. MR**0163331****[4]**Marston Morse,*The calculus of variations in the large*, American Mathematical Society Colloquium Publications, vol. 18, American Mathematical Society, Providence, RI, 1996. Reprint of the 1932 original. MR**1451874****[5]**Robert Hermann,*Differential geometry and the calculus of variations*, Mathematics in Science and Engineering, Vol. 49, Academic Press, New York-London, 1968. MR**0233313**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1974-0334247-9

Keywords:
Periodic transformation,
manifold,
fixed point set,
isometry,
Lefschetz number,
Betti number,
Morse function,
coherency,
fixed point theorem

Article copyright:
© Copyright 1974
American Mathematical Society