The fixed-point property of -connected -manifolds

Author:
S. Y. Husseini

Journal:
Trans. Amer. Math. Soc. **220** (1976), 343-359

MSC:
Primary 55C20; Secondary 57D99

DOI:
https://doi.org/10.1090/S0002-9947-1976-0420601-5

MathSciNet review:
0420601

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Abstract: Suppose that *M* is a -connected smooth and compact manifold of dimension 4*m*. Assume that its intersection pairing is positive definite, and denote its signature by . Two notions are introduced. The first is that of a -map where and an integer. It describes the concept of *f* preserving up to multiplication by outside a point. The second notion is that of being sufficiently asymmetric. It describes in terms of the Chern class of the concept that the restrictions of to the 2*m*-spheres realizing a basis for are sufficiently different so that no map which preserves can move the spheres among themselves. One proves that -maps with being sufficiently asymmetric have fixed points, except possibly when . On taking 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 -maps as well as the question of the preservation of the fixed-point property under products are also discussed.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1976-0420601-5

Keywords:
Smooth manifold,
intersection pairing,
signature,
*K*-theory,
Chern classes,
fixed-point property,
Lefschetz number

Article copyright:
© Copyright 1976
American Mathematical Society