Proceedings of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6826(e) ISSN 0002-9939(p)

Fixed points and powers of self-maps of $H$ -spaces

Author(s): Gregory Lupton; John Oprea
Journal: Proc. Amer. Math. Soc. 124 (1996), 3235-3239.
MSC (1991): Primary 55M20, 55P45, 55P62
MathSciNet review: 1328360
Retrieve article in: PDF
This article is available free of charge

Abstract | References | Similar articles | Additional information

Abstract: We give a new proof of the following result due to Duan: Let $f\,:\, X \to X$ be any self-map of a finite connected $H$ -space. Then $f^{k}$ has a fixed point if $k \geq 2$ . Our proof is based on an approach due to Steve Halperin, whereby the Lefschetz number of a self-map is expressed in terms of the eigenvalues of the induced homomorphism of rational homotopy groups. This allows us to give a considerably shorter proof which avoids most of the technicalities of the original proof.

References:

[Bro]: R. F. Brown, The Lefschetz Fixed Point Theorem, Scott, Foresman and Co., 1971. MR 44:1023
[G-M]: P. Griffiths and J. Morgan, Rational Homotopy Theory and Differential Forms, Birkhäuser, 1981. MR 82m:55014
[Dua-1]: Duan H., The Lefschetz Number of Self-Maps of Lie Groups, Proc. A. M. S. 104 (1988), 1284--1286. MR 89d:55002
[Dua-2]: Duan H., A characteristic polynomial for self-maps of -spaces, Quart. J. Math. Oxford 44 (2) (1993), 315--325. MR 94j:55003
[Hal]: S. Halperin, Spaces whose rational homology and de Rham homotopy are both finite dimensional, Astérisque: Homotopie algèbrique et algébra locale 113-114 (1984), Soc. Math. France, 198--205. MR 86a:55014
[Hun]: T. Hungerford, Abstract Algebra, An Introduction, Saunders, 1990.
[Lan]: S. Lang, Algebra, corrected reprint, Addison Wesley, 1971. MR 33:5416
[Spa]: E. Spanier, Algebraic Topology, McGraw-Hill, 1966. MR 35:1007

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|