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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The fixed-point property of $ (2m-1)$-connected $ 4m$-manifolds

Author: S. Y. Husseini
Journal: Trans. Amer. Math. Soc. 220 (1976), 343-359
MSC: Primary 55C20; Secondary 57D99
MathSciNet review: 0420601
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Abstract: Suppose that M is a $ (2m - 1)$-connected smooth and compact manifold of dimension 4m. Assume that its intersection pairing is positive definite, and denote its signature by $ \sigma $. Two notions are introduced. The first is that of a $ (\xi ,\lambda )$-map $ f:M \to M$ where $ \xi \in K(M)$ and $ \lambda $ an integer. It describes the concept of f preserving $ \xi $ up to multiplication by $ \lambda $ outside a point. The second notion is that of $ \xi $ being sufficiently asymmetric. It describes in terms of the Chern class of $ \xi $ the concept that the restrictions of $ \xi $ to the 2m-spheres realizing a basis for $ {H_{2m}}(M;Z)$ are sufficiently different so that no map which preserves $ \xi $ can move the spheres among themselves. One proves that $ (\xi ,\lambda )$-maps with $ \xi $ being sufficiently asymmetric have fixed points, except possibly when $ \sigma = 2$. On taking $ \xi $ to be the complexification of the tangent bundle of M, one sees that mainfolds with sufficiently asymmetric tangent structures have the fixed point property with respect to a family of maps which includes diffeomorphisms. The question of the existence of $ (\xi ,\lambda )$-maps as well as the question of the preservation of the fixed-point property under products are also discussed.

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Keywords: Smooth manifold, intersection pairing, signature, K-theory, Chern classes, fixed-point property, Lefschetz number
Article copyright: © Copyright 1976 American Mathematical Society

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