Duan's fixed point theorem: proof and generalization (Q2491510)

Suppose that \(X\), a space of the homotopy type of a finite, connected CW-complex, is a homotopy-associative \(H\)-space with multiplication \(m\). The \textit{power map} \(p_k : X \to X\) is the identity if \(k = 1\) and then, for an integer \(k > 1\), it is defined inductively by \(p_k(x) = m(p_{k-1}(x), x)\). The theorem of the title, due to \textit{H. Duan} [Q. J. Math., Oxf. II. Ser. 44, 315--325 (1993; Zbl 0806.55002)] states that if \(f : X \to X\) is any map, then the Lefschetz number \(L(p_kf)\) is nonzero for any \(k \geq 2\) and therefore \(p_kf\) has a fixed point. Thus, for instance, if \(X\) is a topological group then, for any map \(f : X \to X\), the equation \(f(x)^k = x\) must have a solution if \(k \geq 2\). This paper contains a very clever new proof of Duan's fixed point theorem that is only one page long and is an elementary consequence of the well-known Hopf-Leray-Samelson structure theorem for the rational cohomology algebra \(H^*(X)\) of \(X\). The principle new result in the paper concerns a class of spaces \(X\) whose rational cohomology algebra is considerably more general than that of the spaces of Duan's theorem. It includes all spaces formed via finite cartesian products from homotopy-associative \(H\)-spaces, projective spaces and spheres in which not two of the projective spaces and even-dimensional spheres are of the same dimension. A map \(f : X \to X\) induces linear transformations \(f_V\) and \(f_W\) of the indecomposables of \(H^*(X)\) of odd and even degree, respectively. The main theorem states that if \(-1\) is not an eigenvalue of \(f_W\), then \(L(f) = 0\) if and only if \(1\) is an eigenvalue of \(f_V\). A space \(X\) to which this theorem applies does not necessarily support an \(H\)-space structure, but there is a map \(\mu_\theta : X \to X\), called a \textit{\(\theta\)-structure}, which is a generalization of the power map [\textit{Y. Hemmi, K. Morisugi} and \textit{H. Oshima}, J. Math. Soc. Japan 49, 439--453 (1997; Zbl 0904.55002)]. As a consequence of the main theorem, the author extends Duan's theorem by presenting conditions on a map \(f : X \to X\) and a \(\theta\)-structure \(\mu_\theta\) so that both \(f\mu_\theta\) and \(\mu_\theta f\) have nonzero Lefschetz numbers.