On realizing Nakaoka's coincidence point transfer as an S-map (Q1085482)

Let \(p: E\to B\) be a fiber bundle with compact manifold fiber. The author [J. Differ. Geom. 10, 39-48 (1975; Zbl 0294.55009)] defined a ''transfer'' \(\tau\) : \(H^*(E)\to H^*(B)\) in singular cohomology theory such that \(\tau p^*\) is multiplication by the Euler characteristic of the fiber. \textit{J. C. Becker} and the author [Topology 14, 1-12 (1975; Zbl 0306.55017)] proved that there is a stable map which induces \(\tau\), hence such a transfer exists in every cohomology theory. Not all transfers can be so induced. Given another fiber bundle p': \(E\to B\) over the same base space with fiber a manifold of the same dimension, and given maps f, \(g: E\to E'\) covering the identity map of B, \textit{M. Nakaoka} [J. Math. Soc. Japan 32, 751-779 (1980; Zbl 0447.55001)] defined a transfer in singular cohomology \(\tau\) : \(H^*(E)\to H^*(B)\) such that \(\tau p^*\) is multiplication by the Lefschetz coincidence index of the restrictions of f and g to a fiber. The result of the present paper is that, under reasonable hypotheses, Nakaoka's transfer is induced by a stable map.