Fixed points and powers of self-maps of $H$-spaces
HTML articles powered by AMS MathViewer
- by Gregory Lupton and John Oprea PDF
- Proc. Amer. Math. Soc. 124 (1996), 3235-3239 Request permission
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
- Robert F. Brown, The Lefschetz fixed point theorem, Scott, Foresman & Co., Glenview, Ill.-London, 1971. MR 0283793
- Phillip A. Griffiths and John W. Morgan, Rational homotopy theory and differential forms, Progress in Mathematics, vol. 16, Birkhäuser, Boston, Mass., 1981. MR 641551
- Hai Bao Duan, The Lefschetz number of self-maps of Lie groups, Proc. Amer. Math. Soc. 104 (1988), no. 4, 1284–1286. MR 935107, DOI 10.1090/S0002-9939-1988-0935107-8
- Duan Haibao, A characteristic polynomial for self-maps of $H$-spaces, Quart. J. Math. Oxford Ser. (2) 44 (1993), no. 175, 315–325. MR 1240474, DOI 10.1093/qmath/44.3.315
- Irene Llerena, Sur la localisation des fibrations faiblement nilpotentes, C. R. Acad. Sci. Paris Sér. I Math. 299 (1984), no. 9, 393–396 (French, with English summary). MR 764091
- T. Hungerford, Abstract Algebra, An Introduction, Saunders, 1990.
- Serge Lang, Algebra, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965. MR 0197234
- Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210112
Additional Information
- Gregory Lupton
- Affiliation: Department of Mathematics, Cleveland State University, Cleveland, Ohio 44115
- MR Author ID: 259990
- Email: Lupton@math.csuohio.edu
- John Oprea
- Affiliation: Department of Mathematics, Cleveland State University, Cleveland, Ohio 44115
- MR Author ID: 134075
- Email: Oprea@math.csuohio.edu
- Received by editor(s): October 20, 1994
- Received by editor(s) in revised form: April 3, 1995
- Communicated by: Thomas Goodwillie
- © Copyright 1996 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 124 (1996), 3235-3239
- MSC (1991): Primary 55M20, 55P45, 55P62
- DOI: https://doi.org/10.1090/S0002-9939-96-03405-3
- MathSciNet review: 1328360