Coincidence Nielsen numbers for covering maps for orientable and nonorientable manifolds

Let f, g : M → N be maps between closed manifolds of the same dimension, and let p : M → M and p' : N → N be finite regular covering maps. If the manifolds M and N are orientable, then, under certain conditions, the Nielsen number N (f,g) of f and g can be computed as a linear combination of the Nielsen numbers of the lifts of f and g. In the non-orientable case, using semi-index, we introduce two new Nielsen numbers. The first one is the Linear Nielsen number NL (f,g), which is a linear combination of the Nielsen numbers of the lifts of f and g. The second one is the Non-linear Nielsen number NED (f,g). It is the number of certain essential classes whose inverse images by p are inessential Nielsen classes. In fact, N (f,g) = NL (f,g) + NED (f,g), where by abuse of notation, N (f,g) denotes the coincidence Nielsen number defined using semi-index