Roots of iterates of maps (Q1902995)

In this interesting article the authors are concerned with root classes of a selfmap \(f : X \to X\) of a path-connected topological space \(X\). If \(f^n x = f^n y = a\) then \(x\), \(y\) are said to be in the same root class of \(f^n\) if there exists a path \(p : [0,1] \to X\) from \(x\) to \(y\) such that \(f^n \circ p\) is homotopic to the constant path at \(a\). A root class \(R^n_0\) of an iterated map \(f^n : X \to X\) is said to be iterate-essential if the following holds: whenever there is a homotopy \(H\) between \(f\) and a selfmap \(g := H(\cdot, 1) : X \to X\) then there is a root class \(R^n_1\) of \(g\) and a path \(p\) which starts in \(R^n_0\) and ends in \(R^n_1\) such that \(t \mapsto H^n (p(t),t)\) is homotopic to the constant path at \(a\) where \(H^n\) denotes the homotopy between \(f^n\) and \(g^n\) which is induced by \(H\). A root \(x\) of \(f^n\) is said to be irreducible if \(n\) is the minimal number such that \(f^n x = a\) -- this notion is then extended to root classes (which is much less straightforward than it looks at a first glance). The Nielsen number \(NI_n(f;a)\) of irreducible roots of \(f^n\) at \(a\) is then defined to be the number of iterate-essential and irreducible root classes of \(f^n\). The authors undertake a careful study of the Nielsen numbers including a proof of homotopy invariance. For \(X\) a closed manifold, the authors succeed in explicitly determining the Nielsen number \(NI_n (f;a)\) in terms of the fundamental group of \(X\) and the induced homomorphism \(f_\#\) provided that for all \(x, a \in X\) and all \(n \geq 1\) there is a root of \(f^n\) at \(a\) and every root class of \(f^n\) is essential. If, finally, a closed manifold \(X\) admits an \(H\)-space structure the authors explain how their theory can be applied to the theory of primitive roots of unity.