Nielsen numbers and pullbacks
All spaces are compact, connected ANRs. Given an orientable Hurewicz fiber space \(p: E\to B\) and a fiber-preserving map \(\epsilon\) : \(E\to E\), let \(\beta\) : \(B\to B\) be such that \(\beta p=p\epsilon\). Assuming, just for convenience of explanation, that \(\beta\) has a fixed point b, let \(\epsilon_ b: p^{-1}(b)\to p^{-1}(b)\) be the restriction of \(\epsilon\). The reviewer [Pac. J. Math. 21, 465-472 (1967; Zbl 0169.258)] showed that the Lefschetz numbers are related by a product formula: \(L(\epsilon)=L(\beta)L(\epsilon_ b)\). (The proof required the spaces to be finite polyhedra. An appendix to this paper gives the proof for the ANR case that was discovered independently by M. McCord and E. Fadell but never published.) A corresponding, but necessarily more complicated, product formula for the Nielsen numbers was established by \textit{C. You} [ibid. 100, 217-241 (1982; Zbl 0512.55004)]. Now let \(g: A\to B\) be a map and denote by \(A\Pi\) E the pullback of p over g. Given a map \(\alpha\) : \(A\to A\) such that \(\beta g=g\alpha\), there is a ''pulled back'' map \(\alpha\Pi\epsilon\) : \(A\Pi\) \(E\to A\Pi E\). The authors observe that the product theorem above for the Lefschetz numbers can be used to establish a formula of this type for the pullback, namely, \(L(\alpha \Pi \epsilon)L(\beta)=L(\alpha)L(\epsilon)\). The main result of the paper is a product formula (but, again, necessarily a more complicated one) that relates the Nielsen numbers, thus extending You's result to pullbacks. Geometric consequences of the formula include sufficient conditions for \(\alpha\Pi\epsilon\) to be homotopic to a fixed point free map and, taking \(A=\{b\}\), for information about \(\epsilon_ b\) to imply that \(\epsilon\) itself is homotopic to a fixed point free map. The proof of the Nielsen product formula for pullbacks is based on an interesting, purely algebraic, theory of Reidemeister numbers for pullbacks.
scientific article; zbMATH DE number 4003056
scientific article