When the holonomy group is trivial, [GAMMA] can be considered to be a lattice in G and the corresponding manifold [GAMMA]\G is a nilmanifold. When G is abelian, i.e.
It is known that every map on a nilmanifold is weakly Jiang, due to the result of Anosov () or Fadell and Husseini ().
Therefore, every infranilmanifold of the form [GAMMA]\G is finitely covered by a nilmanifold [conjunction]\G, such that every continuous map f : [GAMMA]\G [right arrow] [GAMMA]\G can be lifted to a map [bar.f] : [conjunction]\G [right arrow] [conjunction]\G.
This actually means that if we choose good representatives in the Reidemeister classes of [[[[beta].sub.r]].sub.r] and [[[[beta].sub.s]].sub.s], p'(Fix([[beta].sub.n][[~.g].sup.n])) will reduce to both p'(Fix([[beta].sub.r][[~.g].sup.r])) and p'(Fix([[beta].sub.s][[~.g].sup.s])) on our nilmanifold [LAMBDA]\G.
[MATHEMATICAL EXPRESSION OMITTED] then M is diffeomorphic to a noncompact nilmanifold
As shown in [1, 2, 7, 8, 3] (see also Theorems 2.1 and 2.2) there are simple formulas for the numbers N([f.sup.m]), N[P.sub.m](f) and N[[PHI].sub.m](f) for fixed m on tori and nilmanifolds. These formulas involve the linearization F of f (see  and section 2.1), which any self map of a torus or nilmanifold possesses.
Of course we are thinking of A as a self map of nilmanifold or torus.
In the first part of this subsection we remind the reader of the concept of linearization, of a self map of a torus or nilmanifold, details can be found in [3,10].
McCord, Lefschetz and Nielsen coincidence numbers on nilmanifolds
and solvmanifolds II, Topology Appl., 75 (1997), 81-?92.
 Edward Keppelmann, Periodics points on nilmanifolds
and solvmanifolds, Pacific Journal of Mathematics, vol.164 (1) (1994), 105-128.
In , flat and nilmanifolds
whose fundamental groups possess property [R.sub.[infinity]] were constructed.