A Nielsen type number for fibre preserving maps

Abstract

In this paper, we introduce a Nielsen type number N F (ƒ, p) for a fibre preserving map ƒ of a fibration p; we show that it is a lower bound for the least number of fixed points within the fibre homotopy class of ƒ. The number N F (ƒ, p), which can be thought of as the dual of the relative Nielsen number due to Schirmer, is often much bigger than the ordinary Nielsen number, N(ƒ), of ƒ. It shares with N(ƒ) such properties as homotopy invariance and commutativity. The definition of N F (ƒ, p) is reminiscent of the so-called naïve product formula due to Brown. In this paper, we also exhibit and exploit a connection between the relative Nielsen number and N F (ƒ, p); we compare N F (ƒ, p) and N(ƒ); give necessary and sufficient conditions for N F (ƒ, p) and N(ƒ) to coincide, and show, under fairly mild conditions, that our lower bound is sharp. Some corollaries concerning minimum fixed point sets for ordinary Nielsen numbers of a fibre map are given.