On the homotopy of closed manifolds and finite CW-complexes

Abstract

We study the finite generation of homotopy groups of closed manifolds and finite CW-complexes by relating it to the cohomology of their fundamental groups. Our main theorems are as follows: when $X$ is a finite CW-complex of dimension $n$ and $\pi_1(X)$ is virtually a Poincar\'e duality group of dimension $\geq n-1$, then $\pi_i(X)$ is not finitely generated for some $i$ unless $X$ is homotopy equivalent to the Eilenberg--MacLane space $K(\pi_1(X),1)$; when $M$ is an $n$-dimensional closed manifold and $\pi_1(M)$ is virtually a Poincar\'e duality group of dimension $\ge n-1$, then for some $i\leq [n/2]$, $\pi_i(M)$ is not finitely generated, unless $M$ itself is an aspherical manifold. These generalize theorems of M. Damian from polycyclic groups to any virtually Poincar\'e duality groups. When $\pi_1(X)$ is not a virtually Poincar\'e duality group, we also obtained similar results. As a by-product we showed that if a group $G$ is of type F and $H^i(G,\mathbb{Z} G)$ is finitely generated for any $i$, then $G$ is a Poincar\'e duality group. This recovers partially a theorem of Farrell.