Wecken's theorem for periodic points (Q1401005)

For \(f:X\to X\) a map of a finite polyhedron, the Nielsen number \(N(f)\) is a lower bound for the number of fixed points of every map \(g\) homotopic to \(f\). In 1942, Wecken proved that, if \(X\) is a manifold of dimension at least three, then the Nielsen number is a sharp lower bound in the sense that there exists a map \(g\) homotopic to \(f\) that has exactly \(N(f)\) fixed points. Given a natural number \(n\), the periodic points of \(f\) of period \(n\) are the fixed points of the \(n\)-th iterate \(f^n\) of \(f:X\to X\). There is a lower bound \(NF_n(f)\), due to Jiang, for the number of periodic points of period \(n\) for every map \(g\) homotopic to \(f\). In 1980, Benjamin Halpern published an abstract claiming that \(NF_n(F)\) is also a sharp lower bound: there is a map \(g\) homotopic to \(f\) that has exactly \(NF_n(f)\) periodic points of period \(n\), provided that \(X\) is a differentiable manifold of dimension at least five. However, no proof was ever published and his claim became known as the ``Halpern conjecture''. This very impressive paper presents a proof of the Halpern conjecture (it actually does more since \(X\) need only be a PL manifold of dimension at least four). The proof makes extensive use of techniques published recently by the same author [Math. Ann. 321, 107-130 (2001; Zbl 0994.55003)]. In that earlier paper, he showed that if \(NF_n(f)=0\) then \(f\) can be homotoped to a map \(g\) such that \(g^n\) is fixed point free. To understand the difficulty of establishing even that case, consider the fact that the map \(f\) must undergo a homotopy that eliminates not only its own fixed points but also, and simultaneously, those of \(f^k\) for every natural number \(k\) that divides \(n\). In the general case proved in this paper, the main issue is not to eliminate all the fixed points of certain iterates but rather to combine them (more accurately, combine their orbits) to reduce their number to the extent possible. Once that number is reduced, it may be possible to eliminate some orbits entirely, using the previously developed techniques. The techniques in this paper, as in the previous one, consist of explicit constructions of geometric topology. The paper is well-organized and the constructions are carefully explained.