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 Poincaré 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 Poincaré 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 [Trans. Amer. Math. Soc. 361 (2009), pp. 1791–1809] from polycyclic groups to any virtually Poincaré duality groups.