Homotopy types and Nielsen numbers of periodic homotopy idempotents on tori (Q499948)

Let \(f:X\to X\) be a continuous self-map of a compact polyhedron, and let \(k\) be a positve integer. \(f\) is a periodic homotopy idempotent of period \(k\), if \(f^r\) is homotopic to \(f^{r+k}\) for some integer \(r\geq 0\). It is a periodic homotopy equivalence if \(f^k\) is homotopic to the identity. The authors determine the Nielsen numbers of periodic homotopy idempotents and equivalences on the \(n\)-dimensional torus \(T^n\). For example, if \(f: T^4\to T^4\) is a periodic equivalence, then \(N(f)\) is \(0,1, 2,3,4,5,6,8,9,12\) or \(16\). If \(f:T^6\to T^6\) is a periodic homotopy idempotent, then \(N(f)\) is \(0,1,3,5,7,9,15\) or \(27\). In the case \(n=2\) the authors classify the homotopy types of periodic idempotents using a number-theoretic discussion.