Group actions on aspherical $A_{k}(N)$-manifolds

Abstract:

By an aspherical ${A_k}(N)$-manifold, we mean a compact connected manifold $M$ together with a map $f$ from $M$ into an aspherical complex $N$ such that ${f^{\ast }}: H^k(N;Q)\to H^k(M;Q)$ is nontrivial. In this paper we shall show that if ${S^1}$ acts effectively and smoothly on a smooth aspherical ${A_k}(N)$-manifold, $k > 1$, $N$ a closed oriented Riemannian $k$-manifold, with strictly negative curvature, and the $K$-degree $K(f) \ne 0$, then the fixed point set $F$ is not empty, and at least one component of $F = { \cup _{j}}{F_j}$ is an aspherical ${A_k}(N)$-manifold. Moreover, ${\operatorname {Sign}}(f) = {\Sigma _j} {\operatorname {Sign}}(f|{F_j})$. We also study the degree of symmetry and semisimple degree of symmetry of aspherical ${A_k}(N)$-manifolds.