A minimum fixed point theorem for smooth fiber preserving maps
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- by Catherine Lee PDF
- Proc. Amer. Math. Soc. 135 (2007), 1547-1549 Request permission
Abstract:
Let $p:E\rightarrow B$ be a smooth fiber bundle. Given a smooth fiber preserving map $f:E\rightarrow E$, we will show that $f$ can be deformed by a smooth, fiber preserving homotopy to a smooth map $g:E\rightarrow E$ such that the number of fixed points of $g$ is equal to the fiberwise Nielsen number of $f$.References
- Robert F. Brown, On a homotopy converse to the Lefschetz fixed point theorem, Pacific J. Math. 17 (1966), 407–411. MR 195083, DOI 10.2140/pjm.1966.17.407
- B. A. Dubrovin, A. T. Fomenko, and S. P. Novikov, Modern geometry—methods and applications. Part II, Graduate Texts in Mathematics, vol. 104, Springer-Verlag, New York, 1985. The geometry and topology of manifolds; Translated from the Russian by Robert G. Burns. MR 807945, DOI 10.1007/978-1-4612-1100-6
- Philip R. Heath, Ed Keppelmann, and Peter N.-S. Wong, Addition formulae for Nielsen numbers and for Nielsen type numbers of fibre preserving maps, Topology Appl. 67 (1995), no. 2, 133–157. MR 1362079, DOI 10.1016/0166-8641(95)00019-8
- Bo Ju Jiang, Fixed point classes from a differential viewpoint, Fixed point theory (Sherbrooke, Que., 1980) Lecture Notes in Math., vol. 886, Springer, Berlin-New York, 1981, pp. 163–170. MR 643005
- C. Lee, The affect of smoothness and derivative conditions on the fixed point sets of smooth maps, Ph.D. Thesis, University of California, Los Angeles, December, 2005.
- Franz Wecken, Fixpunktklassen. III. Mindestzahlen von Fixpunkten, Math. Ann. 118 (1942), 544–577 (German). MR 10281, DOI 10.1007/BF01487386
Additional Information
- Catherine Lee
- Affiliation: Department of Mathematics, University of California, Los Angeles, California 90095-1555
- Address at time of publication: 1111 Laveta Terrace, Los Angeles, California 90026
- Email: cathylee@math.ucla.edu
- Received by editor(s): July 7, 2005
- Received by editor(s) in revised form: December 2, 2005
- Published electronically: November 15, 2006
- Additional Notes: This paper is based on a part of the author’s Ph.D. dissertation written under the supervision of Robert F. Brown.
- Communicated by: Paul Goerss
- © Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 135 (2007), 1547-1549
- MSC (2000): Primary 55M20, 55R10, 58A05
- DOI: https://doi.org/10.1090/S0002-9939-06-08600-X
- MathSciNet review: 2276665
Dedicated: This paper is dedicated to my advisor, Robert F. Brown