VIRTUAL PERMUTATIONS OF Z[Zn] COMPLEXES

We extend the characterization of virtual permutation endomorphisms in the case where IIi(Ai) = Zn. We show that for endomorphisms of Z[Zn] complexes the appropriate eigenvalue condition is that all eigenvalues be roots of units of the group ring Z[Zn\. Among these endomorphisms the virtual permutations are detected by Kq. The main application is in identifying Morse-Smale isotopy classes on these manifolds. Morse-Smale diffeomorphisms are the simplest structurally stable dynamical systems. An important problem is to determine when a given isotopy class contains a Morse-Smale diffeomorphism. This problem was reduced by Shub and Sullivan to the algebraic problem of representing homology maps by special matrices called virtual permutations. In this note we report progress in the nonsimply connected case.