Fixed points and powers of self-maps of -spaces

Authors:
Gregory Lupton and John Oprea

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

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Abstract | References | Similar Articles | Additional Information

Abstract: We give a new proof of the following result due to Duan: Let be any self-map of a finite connected -space. Then has a fixed point if . 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.

**[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**

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Additional Information

**Gregory Lupton**

Affiliation:
Department of Mathematics, Cleveland State University, Cleveland, Ohio 44115

Email:
Lupton@math.csuohio.edu

**John Oprea**

Affiliation:
Department of Mathematics, Cleveland State University, Cleveland, Ohio 44115

Email:
Oprea@math.csuohio.edu

DOI:
https://doi.org/10.1090/S0002-9939-96-03405-3

Keywords:
Fixed point,
Lefschetz number,
$H$-space,
rational homotopy

Received by editor(s):
October 20, 1994

Received by editor(s) in revised form:
April 3, 1995

Communicated by:
Thomas Goodwillie

Article copyright:
© Copyright 1996
American Mathematical Society