Addition formulae for Nielsen numbers and Nielsen type numbers of fibre preserving maps defined locally

摘要

In 1995 Heath, Keppelmann and Wong studied addition formulae for Nielsen numbers and for Nielsen type numbers of fibre preserving maps. In this paper we generalize these ideas to the setting of fibre preserving maps defined locally. Let (E, p, B) be a fibre space with E, B and all fibres compact connected ANR's, V be an open subset in B, and U = p(-1)(V). Let f : U - E be a fibre preserving map, i.e., a map for which there exists a map (f) over bar : V - B with pf = (f) over barp(U,V), where P-U,P-V : U - V is the restriction of p. We give necessary and sufficient conditions for when a naive addition formula expressing the local Nielsen number n(f; U) as a simple sum of Nielsen numbers on the fibres is valid. We also introduce a Nielsen type number n(f) (f; U, p) for a fibre preserving map f : U - E, we show that it is a lower bound for the least number of fixed points within the fibre homotopy class of f. Some calculations and examples are given. (C) 2020 Elsevier B.V. All rights reserved.