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Duan's fixed point theorem: Proof and generalization


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.



  1. 1.

    Brown RF: The Lefschetz Fixed Point Theorem. Scott, Foresman, Illinois; 1971:vi+186.

  2. 2.

    Félix Y, Halperin S, Thomas J-C: Rational Homotopy Theory, Graduate Texts in Mathematics. Volume 205. Springer, New York; 2001:xxxiv+535.

  3. 3.

    Griffiths PA, Morgan JW: Rational Homotopy Theory and Differential Forms, Progress in Mathematics. Volume 16. Birkhäuser, Massachusetts; 1981:xi+242.

  4. 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.

  5. 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.

  6. 6.

    Hemmi Y, Morisugi K, Ooshima H: Self maps of spaces. Journal of the Mathematical Society of Japan 1997,49(3):438–453.

  7. 7.

    Hungerford T: Abstract Algebra: An Introduction. Saunders college, Pennsylvania; 1990.

  8. 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/S0002-9939-96-03405-3

  9. 9.

    Milnor JW, Moore JC: On the structure of Hopf algebras. Annals of Mathematics. Second Series (2) 1965, 81: 211–264. 10.2307/1970615

  10. 10.

    Spanier EH: Algebraic Topology. McGraw-Hill, New York; 1966:xiv+528.

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Correspondence to Martin Arkowitz.

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Arkowitz, M. Duan's fixed point theorem: Proof and generalization. Fixed Point Theory Appl 2006, 17563 (2006).

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