Concentrated cyclic actions of high periodicity

Author:

Berend Daniel,Katz Gabriel

Abstract

The class of concentrated periodic diffeomorphisms g : M M 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 g and the dimension of M M ) arc on the circle. In many ways, the cyclic action generated by such a g 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, Sign ( g , M ) = Sign ( M g ) \operatorname {Sign} (g,M) = \operatorname {Sign} ({M^g}) , provided that the left-hand side is an integer; as for prime power order actions, g g cannot have a single fixed point if M M is closed. A variety of integrality results, relating to the usual signatures of certain characteristic submanifolds of the regular neighbourhood of M g {M^g} in M M to Sign ( g , M ) \operatorname {Sign} (g,M) via the normal g g -representations, is established.

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,General Mathematics

Reference5 articles.

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5. The index of manifolds with toral actions and geometric interpretations of the 𝜎(∞,(𝑆¹,𝑀ⁿ)) invariant of Atiyah and Singer;Kawakubo, Katsuo,1972

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