Fixed points of fiber maps of Montgomery–Samelson fiberings over intervals

A Montgomery–Samelson (MS) fibering is a fibering with singularities such that the singular fibers are points. A fiber map is a map that preserves fibers and takes singular fibers to singular fibers. For MS fiberings that are the suspension of a map of a space to a point, and hence the base space is an interval, we obtain a formula for the minimum number of fixed points among all fiber maps that are fiber homotopic to a fiber map from the fibering to itself. The formula calculates that minimum number in terms of the minimum number of fixed points among the maps homotopic to self-maps of nonsingular fibers that are restrictions of the fiber map. These minimum numbers can often be computed by Nielsen fixed point theory. We present an example of an MS fibering in which the total space and nonsingular fibers are simply-connected and yet, in contrast to fiber maps of fiberings without singularities, for each natural number k there is a fiber map f[k] such that the minimum number of fixed points of the fiber maps fiber homotopic to f[k] is k.