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

 

Concentrated cyclic actions of high periodicity


Authors: Daniel Berend and Gabriel Katz
Journal: Trans. Amer. Math. Soc. 323 (1991), 665-689
MSC: Primary 57S17; Secondary 57R20, 57S15, 58G10
MathSciNet review: 1005074
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Abstract: The class of concentrated periodic diffeomorphisms $ g:M \to M$ is introduced. A diffeomorphism is called concentrated if, roughly speaking, its normal eigenvalues range in a small (with respect to the period of $ g$ and the dimension of $ M$) arc on the circle. In many ways, the cyclic action generated by such a $ g$ behaves on the one hand as a circle action and on the other hand as a generic prime power order cyclic action. For example, as for circle actions, $ \operatorname{Sign} (g,M) = \operatorname{Sign} ({M^g})$, provided that the left-hand side is an integer; as for prime power order actions, $ g$ cannot have a single fixed point if $ M$ is closed. A variety of integrality results, relating to the usual signatures of certain characteristic submanifolds of the regular neighbourhood of $ {M^g}$ in $ M$ to $ \operatorname{Sign} (g,M)$ via the normal $ g$-representations, is established.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1991-1005074-4
PII: S 0002-9947(1991)1005074-4
Article copyright: © Copyright 1991 American Mathematical Society