On the higher homotopy groups of a finite CW-complex
Suppose that \(X\) is a finite CW complex of dimension \(q \geq 2\); this paper gives some results on the higher homotopy groups of \(X\). If the first Betti number \(\beta_{1}(X)\) is large enough, \(>q\) will do, then in the case that \(X\) has an abelian fundamental group, \(\pi_{i}(X)\) is not finitely generated for some \(i = 2, \ldots ,q\). Similar results are obtained when \(\chi(X) \neq 0\) and \(\pi_{1}(X) = {\mathbb Z} \times G\) where \(G\) is finitely generated. In the case where \(\beta_{1}(X)= q \) or \(q-1\) the same result is shown with the proviso that \(X\) does not have the homotopy type of a torus.
scientific article; zbMATH DE number 2173010
scientific article