Open Access

Fixed point indices and manifolds with collars

Fixed Point Theory and Applications20062006:87657

DOI: 10.1155/FPTA/2006/87657

Received: 7 December 2004

Accepted: 24 July 2005

Published: 3 May 2006


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.


Authors’ Affiliations

Mathematics Department, Purdue University


  1. Becker JC, Gottlieb DH: Vector fields and transfers. Manuscripta Mathematica 1991,72(2):111–130.MathSciNetView ArticleMATHGoogle 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.MathSciNetView ArticleMATHGoogle 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)90044-3MathSciNetMATHGoogle 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/BF01214520MathSciNetView ArticleMATHGoogle Scholar
  8. Fadell E: Generalized normal bundles for locally-flat imbeddings. Transactions of the American Mathematical Society 1965, 114: 488–513. 10.1090/S0002-9947-1965-0179795-4MathSciNetView ArticleMATHGoogle 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.View ArticleGoogle 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., BrasilMathSciNetMATHGoogle 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.View ArticleGoogle Scholar
  13. Hopf H: Abbildungsklassen -dimensionaler {M}annigfaltigkeiten. Mathematische Annalen 1927,96(1):209–224. 10.1007/BF01209163MathSciNetView ArticleMATHGoogle Scholar
  14. Hu ST: Fibrings of enveloping spaces. Proceedings of the London Mathematical Society. Third Series 1961, 11: 691–707. 10.1112/plms/s3-11.1.691MathSciNetView ArticleMATHGoogle Scholar
  15. Morse M: Singular points of vector fields under general boundary conditions. American Journal of Mathematics 1929,51(2):165–178. 10.2307/2370703MathSciNetView ArticleMATHGoogle 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.41.5.320MathSciNetView ArticleMATHGoogle 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)90002-5MathSciNetMATHGoogle Scholar


© C.-F. Benjamin and D. H. Gottlieb. 2006

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