Coincidence Nielsen numbers for covering maps for smooth manifolds

Abstract

Let f , g : M → N be maps between closed smooth manifolds of the same dimension, and let p : M ˜ → M and p ′ : N ˜ → N be finite regular covering maps. If the manifolds are nonorientable, using semi-index, we introduce two new Nielsen numbers. The first one is the Linear Nielsen number N L ( f , g ) , which is a linear combination of the Nielsen numbers of the lifts of f and g. The second one is the Nonlinear Nielsen number N ED ( f , g ) . It is the number of certain essential classes whose inverse images by p are inessential Nielsen classes. In fact, N ( f , g ) = N L ( f , g ) + N ED ( f , g ) , where by abuse of notation, N ( f , g ) denotes the coincidence Nielsen number defined using semi-index.