Homotopy type through homology groups (Q6120035)

In this paper the authors consider the homotopy type of the simply connected CW-complex \(X\) satisfying the following condition \((*)\): \[ \tilde{H}_q(X;\mathbb{Z})\cong \begin{cases} \mathbb{Z}^a & \text{ for }q=d \\ \mathbb{Z}^b & \text{ for }q=d+k \\ 0 & \text{ for }q\not=d,d+k, \end{cases} \tag{\(*\)} \] where \(a,b,d,k\) are positive integers with \(d>k\). In particular, they prove that there is a homotopy equivalence \[ X\simeq \big(\bigvee^aS^d\big)\vee \big(\bigvee^bS^{d+k}\big) \] when \(k\in \{1,5,6,13,62\}\). Their proof is based on the computation of the Serre spectral sequence and The Blakers-Massey theorem.