Skip to main content

Fixed point indices and manifolds with collars

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.

[1234567891011121314151617]

References

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

    MathSciNet  Article  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.

    MathSciNet  Article  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

    MathSciNet  Article  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

    MathSciNet  Article  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

    MathSciNet  Article  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

    MathSciNet  Article  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

    MathSciNet  Article  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

    MathSciNet  Article  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 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Daniel Henry Gottlieb.

Rights and permissions

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.

Reprints and Permissions

About this article

Cite this article

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

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

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

Keywords

  • Differential Geometry
  • Computational Biology
  • Point Index