The absolute degree and Nielsen root number of a fibre-preserving map

Abstract

Let p :M→X and q :N→Y be locally trivial bundles, with fibres F and G, respectively, where all the spaces are connected closed manifolds, but neither the manifolds nor the bundles need be orientable. Assume further that dim X=dim Y and dim F=dim G so dim M=dim N. A fibre-preserving map f :M→N induces a map f ̄ :X→Y of the base and maps f x :F x→G f ̄ (x) of the fibres. The purpose of this paper is to relate the Nielsen root number NR( f) of f with that of f ̄ and f x and to do the same for the absolute degree A(f) . If both f ̄ and f x are orientable maps or if f is both orientable and root-essential (that is, if NR( f)>0), then the multiplicative property A(f)= A( f ̄ )· A(f x) is shown to be valid. Applying this property to selfmaps of compact solvmanifolds produces a computational result for the absolute degree of such a map. If f is a root-essential nonorientable map, the multiplicative property for the absolute degree must be modified in a way that includes a factor κ( f) that describes a relationship between the root class structure of f x and that of f. The Nielsen root numbers of any root-essential fibre-preserving map are found to satisfy κ(f)·NR(f)=NR( f ̄ )·NR(f x) . A fibre-preserving version FG(f) of the classical geometric degree G(f) is defined and it is shown that FG(f)= G(f) if and only if the absolute degree has the multiplicative property. Letting MR[ f] denote the minimum number of points in the pre-image of a given point of N among all maps homotopic to f and FMR[ f] the same with regard to fibre-preserving maps and homotopies, every fibre-preserving map satisfies FMR[f]=MR[f x]·MR[ f ̄ ] . Moreover, if none of the manifolds M, X and F are two-dimensional, then FMR[ f]= MR[ f] if and only if NR(f)=NR( f ̄ )·NR(f x) . A new bundle and pairing, the fibrewise orientation bundle and the orientation bundle pairing, are introduced in order to relate the orientation bundles of base, fibre, and total space of a locally trivial bundle.