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  • Research Article
  • Open Access

Duan's fixed point theorem: Proof and generalization

Fixed Point Theory and Applications20062006:17563

  • Received: 25 July 2004
  • Accepted: 21 July 2005
  • Published:


Let be an H-space of the homotopy type of a connected, finite CW-complex, any map and the th power map. Duan proved that has a fixed point if . We give a new, short and elementary proof of this. We then use rational homotopy to generalize to spaces whose rational cohomology is the tensor product of an exterior algebra on odd dimensional generators with the tensor product of truncated polynomial algebras on even dimensional generators. The role of the power map is played by a -structure as defined by Hemmi-Morisugi-Ooshima. The conclusion is that and each has a fixed point.


  • Point Theorem
  • Differential Geometry
  • Fixed Point Theorem
  • Computational Biology


Authors’ Affiliations

Department of Mathematics, Dartmouth College, Hanover, NH 03755, USA


  1. Brown RF: The Lefschetz Fixed Point Theorem. Scott, Foresman, Illinois; 1971:vi+186.Google Scholar
  2. Félix Y, Halperin S, Thomas J-C: Rational Homotopy Theory, Graduate Texts in Mathematics. Volume 205. Springer, New York; 2001:xxxiv+535.View ArticleGoogle Scholar
  3. Griffiths PA, Morgan JW: Rational Homotopy Theory and Differential Forms, Progress in Mathematics. Volume 16. Birkhäuser, Massachusetts; 1981:xi+242.Google Scholar
  4. Haibao D: A characteristic polynomial for self-maps of -spaces. The Quarterly Journal of Mathematics. Oxford. Second Series (2) 1993,44(175):315–325.MathSciNetView ArticleMATHGoogle Scholar
  5. Halperin S: Spaces whose rational homology and de Rham homotopy are both finite-dimensional. In Algebraic Homotopy and Local Algebra (Luminy, 1982), Astérisque. Volume 113–114. Soc. Math. France, Paris; 1984:198–205.Google Scholar
  6. Hemmi Y, Morisugi K, Ooshima H: Self maps of spaces. Journal of the Mathematical Society of Japan 1997,49(3):438–453.MathSciNetView ArticleGoogle Scholar
  7. Hungerford T: Abstract Algebra: An Introduction. Saunders college, Pennsylvania; 1990.Google Scholar
  8. Lupton G, Oprea J: Fixed points and powers of self-maps of -spaces. Proceedings of the American Mathematical Society 1996,124(10):3235–3239. 10.1090/S0002-9939-96-03405-3MathSciNetView ArticleMATHGoogle Scholar
  9. Milnor JW, Moore JC: On the structure of Hopf algebras. Annals of Mathematics. Second Series (2) 1965, 81: 211–264. 10.2307/1970615MathSciNetView ArticleMATHGoogle Scholar
  10. Spanier EH: Algebraic Topology. McGraw-Hill, New York; 1966:xiv+528.MATHGoogle Scholar


© Arkowitz 2006

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.