Lee, Catherine A minimum fixed point theorem for smooth fiber preserving maps. (English) Zbl 1110.55001 Proc. Am. Math. Soc. 135, No. 5, 1547-1549 (2007). Let \(P\colon E\to B\) be a fibration, and \(f\) be a fiber preserving map which induces a map \(\bar f: B\to B\). The fiberwise Nielsen number \(N_{\mathcal{F}}(f, p)\) is defined to be the sum \(\sum _{x\in \xi} N(f_x)\), where \(\xi\) is a set consisting of one point from each essential fixed point class of \(\bar f\) and \(f_x\) is the restriction of \(f\) to the fiber \(p^{-1}(x)\) over \(x\). \(N_{\mathcal{F}}(f, p)\) provides a lower bound for the numbers of fixed points of fiber preserving maps which are homotopic to \(f\) by fiber preserving homotopy.P. Heath, E. Keppelmann and P. Wong [Topology Appl. 67, No. 2, 133–157 (1995; Zbl 0845.55004)] proved that under some assumptions on the fiber space \(E\) and the given fiber preserving map \(f\), the map \(f\) is fiber homotopic to a fiber preserving map \(g\) with exactly \(N_{\mathcal{F}}(f, p)\) fixed points. The author of this paper obtains a similar result in the smooth category. That is, if \(P\colon E\to B\) is a fiber bundle with \(\dim B, F\geq 3\), then any smooth fiber preserving map \(f\) is smoothly fiber preserving homotopic to a map with \(N_{\mathcal{F}}(f, p)\) fixed points. Reviewer: Xuezhi Zhao (Beijing) MSC: 55M20 Fixed points and coincidences in algebraic topology 55R10 Fiber bundles in algebraic topology 58A05 Differentiable manifolds, foundations Keywords:Nielsen number; fiber preserving map; fibration; fiber bundle Citations:Zbl 0845.55004 PDFBibTeX XMLCite \textit{C. Lee}, Proc. Am. Math. Soc. 135, No. 5, 1547--1549 (2007; Zbl 1110.55001) Full Text: DOI References: [1] Robert F. Brown, On a homotopy converse to the Lefschetz fixed point theorem, Pacific J. Math. 17 (1966), 407 – 411. · Zbl 0141.20803 [2] 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. · Zbl 0565.57001 [3] 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. · Zbl 0845.55004 · doi:10.1016/0166-8641(95)00019-8 [4] 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. [5] 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. [6] Franz Wecken, Fixpunktklassen. III. Mindestzahlen von Fixpunkten, Math. Ann. 118 (1942), 544 – 577 (German). · Zbl 0027.26503 · doi:10.1007/BF01487386 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.