Nielsen type numbers and homotopy minimal periods for maps on the 3-nilmanifolds (Q931528)

Let \(M=G/\Gamma\) be a \(3\)-dimensional compact nilmanifold where \(G\) is a connected simply connected nilpotent Lie group and \(\Gamma\) a co-compact lattice. Since \(M\) is a \(K(\Gamma,1)\), every map \(f:M\to M\) is determined, up to homotopy, by the homomorphism \(\varphi:\Gamma \to \Gamma\) induced on \(\Gamma\), which in turn can be extended uniquely to a homomorphism \(\varphi:G\to G\). The characteristic polynomial of the differential \(d\varphi\) on the Lie algebra \(\mathfrak G\) of \(G\) is determined by a \(2 \times 2\) submatrix \(A_{\varphi}\). The Lefschetz number and the Nielsen number of the \(n\)-th iterate \(f^n\) can be explicitly expressed in terms of the eigenvalues of \(A_{\varphi}\). Combining similar calculations of the Nielsen type numbers for periodic points \(NP_n(f)\) and \(N\Phi_n(f)\) by \textit{P. Heath} and \textit{E. Keppelmann} [Topology Appl. 76, No.~3, 217--247 (1997; Zbl 0881.55002)] for \(3\)-dimensional solvmanifolds, the authors give a complete calculation of the two Nielsen type numbers for \(3\)-nilmanifolds in terms of the eigenvalues of \(A_{\varphi}\). They also determine the set of homotopy minimal periods \(\text{HPer}(f)\) of \(f\), thereby correcting the calculation of one case first computed by \textit{J. Jezierski} and \textit{W. Marzantowicz} in [Pac. J. Math. 209, No.~1, 85--101 (2003; Zbl 1049.37033)].