Virtual permutations of $\textbf {Z}[\textbf {Z}^{n}]$ complexes

Abstract:

We extend the characterization of virtual permutation endomorphisms in the case where ${\prod _1}(M) = {Z^n}$. We show that for endomorphisms of $Z[{Z^n}]$ complexes the appropriate eigenvalue condition is that all eigenvalues be roots of units of the group ring $Z[{Z^n}]$. Among these endomorphisms the virtual permutations are detected by ${K_0}$. The main application is in identifying Morse-Smale isotopy classes on these manifolds.