On changing fixed points and coincidences to roots
HTML articles powered by AMS MathViewer
- by Robin Brooks and Peter Wong PDF
- Proc. Amer. Math. Soc. 115 (1992), 527-533 Request permission
Correction: Proc. Amer. Math. Soc. 118 (1993), 1353-1354.
Abstract:
The coincidence problem, finding solutions to $f(x) = g(x)$, can sometimes be converted to a root problem, finding solutions to $\sigma (x) = a$. As an application, we show that for any two maps $f,g:M \to M,N(f,g) = |L(f,g)|$ where $M$ is a compact connected nilmanifold, $N(f,g)$ and $L(f,g)$ are the Nielsen and Lefschetz numbers, respectively, of $f$ and $g$. This result in the case where $g$ is the identity is due to D. Anosov.References
- D. Anosov, The Nielsen number of maps of nilmanifolds, Russian Math. Surveys 40 (1985), 149-150.
- R. Brooks, Certain subgroups of the fundamental group and the number of roots of $f(x) = a$, Amer. J. Math. 95 (1973), 720–728. MR 346777, DOI 10.2307/2373695 —, Coincidences roots and fixed points, PhD Thesis, Univ. of California, Los Angeles, CA, 1967.
- Robin B. S. Brooks and Robert F. Brown, A lower bound for the $\Delta$-Nielsen number, Trans. Amer. Math. Soc. 143 (1969), 555–564. MR 276959, DOI 10.1090/S0002-9947-1969-0276959-X
- Robert F. Brown, The Lefschetz fixed point theorem, Scott, Foresman & Co., Glenview, Ill.-London, 1971. MR 0283793
- A. Dold, Lectures on algebraic topology, Die Grundlehren der mathematischen Wissenschaften, Band 200, Springer-Verlag, New York-Berlin, 1972 (German). MR 0415602
- E. Fadell, Two vignettes in fixed point theory, Topological fixed point theory and applications (Tianjin, 1988) Lecture Notes in Math., vol. 1411, Springer, Berlin, 1989, pp. 46–51. MR 1031781, DOI 10.1007/BFb0086439
- Edward Fadell and Sufian Husseini, On a theorem of Anosov on Nielsen numbers for nilmanifolds, Nonlinear functional analysis and its applications (Maratea, 1985) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 173, Reidel, Dordrecht, 1986, pp. 47–53. MR 852569
- Jerzy Jezierski, The Nielsen number product formula for coincidences, Fund. Math. 134 (1990), no. 3, 183–212. MR 1071665, DOI 10.4064/fm-134-3-183-212
- Bo Ju Jiang, Lectures on Nielsen fixed point theory, Contemporary Mathematics, vol. 14, American Mathematical Society, Providence, R.I., 1983. MR 685755
- Kiang Tsai-han, The theory of fixed point classes, Translated from the second Chinese edition, Springer-Verlag, Berlin; Science Press Beijing, Beijing, 1989. MR 1002187, DOI 10.1007/978-3-642-68133-2
- Christopher K. McCord, Lefschetz and Nielsen coincidence numbers on nilmanifolds and solvmanifolds, Topology Appl. 43 (1992), no. 3, 249–261. MR 1158871, DOI 10.1016/0166-8641(92)90160-2
- Minoru Nakaoka, Coincidence Lefschetz number for a pair of fibre preserving maps, J. Math. Soc. Japan 32 (1980), no. 4, 751–779. MR 589111, DOI 10.2969/jmsj/03240751
- Helga Schirmer, Mindestzahlen von Koinzidenzpunkten, J. Reine Angew. Math. 194 (1955), 21–39 (German). MR 73172, DOI 10.1515/crll.1955.194.21
- James W. Vick, Homology theory, Pure and Applied Mathematics, Vol. 53, Academic Press, New York-London, 1973. An introduction to algebraic topology. MR 0375279
Additional Information
- © Copyright 1992 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 115 (1992), 527-533
- MSC: Primary 55M20
- DOI: https://doi.org/10.1090/S0002-9939-1992-1098397-0
- MathSciNet review: 1098397