Cancelling periodic points (Q5949433)

Let \(M\) be a complete PL manifold and \(f: M\to M\) a map. The Nielsen number \(N(f)\) is a lower bound for the number of fixed points of every map homotopic to \(f\). If the dimension of \(M\) is at least three, then there is a map \(g\) homotopic to \(f\) such that \(g\) has exactly \(N(f)\) fixed points. For a natural number \(n\), the Nielsen periodic point number \(NF_n(f)\) is a lower bound for the number of periodic points of period \(n\) for every map \(g\) homotopic to \(f\). In 1983, Benjamin Halpern claimed that, if \(M\) is closed and of dimension at least five, then there is a map \(g\) homotopic to \(f\) such that \(g\) has exactly \(NF_n(f)\) periodic points of period \(n\). Since no proof of the claim was ever published, it became known as the Halpern Conjecture. This paper considers the special case that \(NF_n(f)= 0\) and proves that, if the dimension of \(M\) is at least four, then it follows that there is a map \(g\) homotopic to \(f\) such that \(g\) has no periodic points of period \(n\). Suppose \(k\) is a divisor of \(n\) and that \(f^\ell\) has no fixed points for all \(\ell\) dividing \(n\) and \(\ell< k\). The problem is to homotope \(f\) to eliminate the fixed points of \(f^k\) in such a way that no periodic points of order \(\ell\) are introduced by the homotopy. Compactness implies that, since \(f^\ell\) has no fixed points, each point of \(M\) is moved at least a certain amount by \(f^\ell\). So, if a deformation of \(f\) to remove the fixed points of \(f^k\) moves points a sufficiently small amount, no periodic points of order \(\ell\) will be introduced. The set of fixed points of \(f^k\) splits into orbits of \(k\) elements and the fact that \(NF_n(f)= 0\) implies \(N(f^k)= 0\) is used to pair up the orbits and then cancel them. The task of constructing the required deformation of \(f\) in this way is an extremely delicate one. The author presents an ingenious sequence of steps, supported by careful arguments, to carry out the construction. In the process, he introduces several techniques that he was able to use subsequently to obtain a complete verification of the Halpern Conjecture.