The fixed-point property of $(2m-1)$-connected $4m$-manifolds
HTML articles powered by AMS MathViewer
- by S. Y. Husseini PDF
- Trans. Amer. Math. Soc. 220 (1976), 343-359 Request permission
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.References
- J. F. Adams, On the groups $J(X)$. IV, Topology 5 (1966), 21–71. MR 198470, DOI 10.1016/0040-9383(66)90004-8
- N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1337, Hermann, Paris, 1968 (French). MR 0240238
- Edward Fadell, Recent results in the fixed point theory of continuous maps, Bull. Amer. Math. Soc. 76 (1970), 10–29. MR 271935, DOI 10.1090/S0002-9904-1970-12358-8
- P. J. Hilton, An introduction to homotopy theory, Cambridge Tracts in Mathematics and Mathematical Physics, No. 43, Cambridge, at the University Press, 1953. MR 0056289
- S. Y. Husseini, Manifolds with the fixed point property. I, Bull. Amer. Math. Soc. 80 (1974), 664–666. MR 343302, DOI 10.1090/S0002-9904-1974-13538-X
- Michel A. Kervaire, Some nonstable homotopy groups of Lie groups, Illinois J. Math. 4 (1960), 161–169. MR 113237
- Hans-Volker Niemeier, Definite quadratische Formen der Dimension $24$ und Diskriminante $1$, J. Number Theory 5 (1973), 142–178 (German, with English summary). MR 316384, DOI 10.1016/0022-314X(73)90068-1
- S. P. Novikov, On manifolds with free abelian fundamental group and their application, Izv. Akad. Nauk SSSR Ser. Mat. 30 (1966), 207–246 (Russian). MR 0196765
- Jean-Pierre Serre, Cours d’arithmétique, Collection SUP: “Le Mathématicien”, vol. 2, Presses Universitaires de France, Paris, 1970 (French). MR 0255476
- C. T. C. Wall, Classification of $(n-1)$-connected $2n$-manifolds, Ann. of Math. (2) 75 (1962), 163–189. MR 145540, DOI 10.2307/1970425
- C. T. C. Wall, Classification problems in differential topology. II. Diffeomorphisms of handlebodies, Topology 2 (1963), 263–272. MR 156354, DOI 10.1016/0040-9383(63)90009-0
Additional Information
- © Copyright 1976 American Mathematical Society
- 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