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 X is a finite CW-complex of dimension n n and π 1 ( X ) \pi _1(X) is virtually a Poincaré duality group of dimension ≥ n − 1 \geq n-1 , then π i ( X ) \pi _i(X) is not finitely generated for some i i unless X X is homotopy equivalent to the Eilenberg–MacLane space K ( π 1 ( X ) , 1 ) K(\pi _1(X),1) ; when M M is an n n -dimensional closed manifold and π 1 ( M ) \pi _1(M) is virtually a Poincaré duality group of dimension ≥ n − 1 \ge n-1 , then for some i ≤ [ n / 2 ] i\leq [n/2] , π i ( M ) \pi _i(M) is not finitely generated, unless M 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.