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Fixed point indices and manifolds with collars

Fixed Point Theory and Applications volume 2006, Article number: 87657 (2006) Cite this article

Abstract

This paper concerns a formula which relates the Lefschetz number for a map to the fixed point index summed with the fixed point index of a derived map on part of the boundary of . Here is a compact manifold and is with a collar attached.

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References

  1. Becker JC, Gottlieb DH: Vector fields and transfers. Manuscripta Mathematica 1991,72(2):111–130.

    Article  MathSciNet  MATH  Google Scholar 

  2. Benjamin C-F: Fixed point indices, transfers and path fields, M.S. thesis. Purdue University, Indiana; 1990.

    Google Scholar 

  3. Brown RF: Path fields on manifolds. Transactions of the American Mathematical Society 1965, 118: 180–191.

    Article  MathSciNet  MATH  Google Scholar 

  4. Brown RF: The {L}efschetz Fixed Point Theorem. Scott, Foresman, Illinois; 1971:vi+186.

    Google Scholar 

  5. Dold A: Fixed point index and fixed point theorem for {E}uclidean neighborhood retracts. Topology. An International Journal of Mathematics 1965, 4: 1–8. 10.1016/0040-9383(65)90044-3

    MathSciNet  MATH  Google Scholar 

  6. Dold A: Lectures on Algebraic Topology, Die Grundlehren der mathematischen Wissenschaften. Volume 200. Springer, New York; 1972:xi+377.

    Google Scholar 

  7. Dold A: The fixed point transfer of fibre-preserving maps. Mathematische Zeitschrift 1976,148(3):215–244. 10.1007/BF01214520

    Article  MathSciNet  MATH  Google Scholar 

  8. Fadell E: Generalized normal bundles for locally-flat imbeddings. Transactions of the American Mathematical Society 1965, 114: 488–513. 10.1090/S0002-9947-1965-0179795-4

    Article  MathSciNet  MATH  Google Scholar 

  9. Gottlieb DH: A de {M}oivre like formula for fixed point theory. In Fixed Point Theory and Its Applications (Berkeley, CA, 1986), Contemp. Math.. Volume 72. Edited by: Brown RF. American Mathematical Society, Rhode Island; 1988:99–105.

    Chapter  Google Scholar 

  10. Gottlieb DH: A de Moivre formula for fixed point theory. ATAS de 5∘ Encontro Brasiliero de Topologia 1988, 53: 59–67. Universidade de Sao Paulo, Sao Carlos S.~P., Brasil

    MathSciNet  MATH  Google Scholar 

  11. Gottlieb DH: On the index of pullback vector fields. In Differential Topology (Siegen, 1987), Lecture Notes in Math.. Volume 1350. Edited by: Koschorke U. Springer, Berlin; 1988:167–170.

    Google Scholar 

  12. Gottlieb DH: Zeroes of pullback vector fields and fixed point theory for bodies. In Algebraic Topology (Evanston, IL, 1988), Contemp. Math.. Volume 96. American Mathematical Society, Rhode Island; 1989:163–180.

    Chapter  Google Scholar 

  13. Hopf H: Abbildungsklassen -dimensionaler {M}annigfaltigkeiten. Mathematische Annalen 1927,96(1):209–224. 10.1007/BF01209163

    Article  MathSciNet  MATH  Google Scholar 

  14. Hu ST: Fibrings of enveloping spaces. Proceedings of the London Mathematical Society. Third Series 1961, 11: 691–707. 10.1112/plms/s3-11.1.691

    Article  MathSciNet  MATH  Google Scholar 

  15. Morse M: Singular points of vector fields under general boundary conditions. American Journal of Mathematics 1929,51(2):165–178. 10.2307/2370703

    Article  MathSciNet  MATH  Google Scholar 

  16. Nash J: A path space and the {S}tiefel-{W}hitney classes. Proceedings of the National Academy of Sciences of the United States of America 1955, 41: 320–321. 10.1073/pnas.41.5.320

    Article  MathSciNet  MATH  Google Scholar 

  17. Pugh CC: A generalized {P}oincaré index formula. Topology. An International Journal of Mathematics 1968, 7: 217–226. 10.1016/0040-9383(68)90002-5

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. 705 Sugar Hill Drive, West Lafayette, IN, 47906, USA

    Chen-Farng Benjamin

  2. Mathematics Department, Purdue University, West Lafayette, IN, 47907, USA

    Daniel Henry Gottlieb

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  1. Chen-Farng Benjamin
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  2. Daniel Henry Gottlieb
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Correspondence to Daniel Henry Gottlieb.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Benjamin, CF., Gottlieb, D.H. Fixed point indices and manifolds with collars. Fixed Point Theory Appl 2006, 87657 (2006). https://doi.org/10.1155/FPTA/2006/87657

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  • DOI: https://doi.org/10.1155/FPTA/2006/87657

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