| Existence theorems are often proved by converting the problems into fixed point problems. And in many cases, besides the existence of solutions, it is useful to know how many solutions there are or at least know a lower bound for their number. The Nielsen theory produces a lower bound N(f) of the number of fixed points of a self map f or the number of coincidence points of a pair of maps (f,g). However, it turns out that the computation of these numbers are not easy to compute. Several works are available in the literature which give theoretical formulae for the Nielsen numbers on particular class of spaces. For example, on a torus, the Nielsen number is obtained by computing the determinant of the homomorphism induced by the maps minus the identity. Some of these works are the starting point of this paper. We restrict ourselves to the case where the space are nilmanifolds and solvmanifolds, which are respectively homogeneous spaces of nilpotent and solvable Lie groups. The goal to establish computable formulae and to write a computer program which computes the Nielsen number, but only requires a minimal amount of information about the spaces and the maps. First, it is shown that a nilmanifold can be decomposed into tori via a tower of fibrations. Then using the Mostow fibration, a solvmanifold can be decomposed into tori as well. The Nielsen theory applied to fibration allows to establish relations between the Nielsen number of the maps on the spaces and those induced on the tori in the decompositions. Thus, all the computations are reduced to calculating Nielsen numbers on tori, and hence calculating determinant of square matrices. These results are implemented in a MACSYMA program, which requires only square matrices as inputs, and produces the Nielsen number (or Nielsen coincidence numbers) as its output. |
