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

Lupton, Gregory; Oprea, John

doi:10.1090/S0002-9939-96-03405-3

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