2018/12/09 by Su, Yang, Wu, Xiaolei
#13D07 #55Q99 #55U30 #Algebraic Topology (math.AT) #FOS: Mathematics #Group Theory (math.GR)
paper · doi:10.48550/arxiv.1812.03452
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 π1(X) is virtually a Poincaré duality group of dimension ≥ n-1, then πi(X) is not finitely generated for some i unless X is homotopy equivalent to the Eilenberg--MacLane space K(π1(X),1); when M is an n-dimensional closed manifold and π1(M) is virtually a Poincaré duality group of dimension ≥ n-1, then for some i≤ [n/2], πi(M) is not finitely generated, unless M itself is an aspherical manifold. These generalize theorems of M. Damian from polycyclic groups to any virtually Poincaré duality groups. When π1(X) is not a virtually Poincaré duality group, we also obtained similar results. As a by-product we showed that if a group G is of type F and Hi(G,ℤ G) is finitely generated for any i, then G is a Poincaré duality group. This recovers partially a theorem of Farrell.