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A minimum fixed point theorem for smooth fiber preserving maps. (English) Zbl 1110.55001

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.

MSC:

55M20 Fixed points and coincidences in algebraic topology
55R10 Fiber bundles in algebraic topology
58A05 Differentiable manifolds, foundations

Citations:

Zbl 0845.55004
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References:

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[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
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