On the higher homotopy groups of a finite CW-complex

Let Q q be a finite connected CW-complex of dimension q ⩾ 2 whose fundamental group is Abelian. Denote by β 1 ( Q ) its first Betti number. ve that if β 1 ( Q ) > q then π i ( Q ) is not finitely generated for some i = 2 , … , q . The same conclusion holds if χ ( Q ) ≠ 0 and π 1 ( Q ) = Z × G where G is finitely presented (but not necessarily Abelian). If π 1 ( Q ) is Abelian and β 1 ( Q ) equals q or q − 1 , we show that π i ( Q ) is not finitely generated for some i = 2 , … , q unless the case when π 1 ( Q ) ∼ Z β 1 ( Q ) and Q has the homotopy type of the β 1 ( Q ) -dimensional torus. s a closed connected manifold, we obtain in the same hypothesis ( β 1 ( Q ) ⩾ q − 1 ) that π i ( Q ) is not finitely generated for some i = 2 , … , max { [ q 2 ] , 3 } except for the case when Q is homotopically equivalent to T q .