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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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  • [AH] John P. Alexander and Gary C. Hamrick, Periodic maps on Poincaré duality spaces, Comment. Math. Helv. 53 (1978), no. 1, 149–159. MR 483537, 10.1007/BF02566071
  • [AS] M. F. Atiyah and I. M. Singer, The index of elliptic operators. III, Ann. of Math. (2) 87 (1968), 546–604. MR 0236952
  • [BK] Daniel Berend and Gabriel Katz, Separating topology and number theory in the Atiyah-Singer 𝑔-signature formula, Duke Math. J. 61 (1990), no. 3, 939–971. MR 1084466, 10.1215/S0012-7094-90-06136-8
  • [H] F. Hirzebruch, Topological methods in algebraic geometry, Third enlarged edition. New appendix and translation from the second German edition by R. L. E. Schwarzenberger, with an additional section by A. Borel. Die Grundlehren der Mathematischen Wissenschaften, Band 131, Springer-Verlag New York, Inc., New York, 1966. MR 0202713
  • [KR] Katsuo Kawakubo and Frank Raymond, The index of manifolds with toral actions and geometric interpretations of the 𝜎(∞,(𝑆¹,𝑀ⁿ)) invariant of Atiyah and Singer, Proceedings of the Second Conference on Compact Transformation Groups (Univ. Massachusetts, Amherst, Mass., 1971) Springer, Berlin, 1972, pp. 228–233. Lecture Notes in Math., Vol. 298. MR 0358840

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Article copyright: © Copyright 1991 American Mathematical Society