Classification of stable homotopy types with torsion-free homology (Q5950614)

Let \({\mathbf F}^k_n\) be the homotopy category consisting of all \((n-1)\)-connected \((n+k)\)-dimensional CW-complexes with finitely generated free homology. If \(k<n-1\), the category \({\mathbf F}^k={\mathbf F}^k_n\) is an additive category which does not depend on \(n\). When \(k=1\) or \(2\), the category \({\mathbf F}^k\) is well known by the earlier works due to J. H. C. Whitehead and S. C. Chang. When \(k=3\) or \(4\), the complete list of indecomposable objects in \({\mathbf F}^k\) is now known by the works of Baues-Hennes and Baues-Drozd. By the above results, the number of indecomposable objects in \({\mathbf F}^k\) is finite if \(1\leq k\leq 4\). So, in this paper, the authors consider the category \({\mathbf F}^k\) when \(k>4\). In particular, they determine the number of indecomposable objects in \({\mathbf F}^k\) explicitly and they also compute the isomorphism class group \(K_0({\mathbf F}^k)\) for all \(k\geq 1\), where \(K_0({\mathbf F}^k)\) denotes the abelian group generated by the objects in \({\mathbf F}^k\) with the relation \(\{X\}+\{Y\}=\{X\vee Y\}\). Moreover, they show that the category \({\mathbf F}^k\) has finite representation type if and only if \(k\leq 5\) and it has wild representation type for \(k\geq 10\).