Let \(n>2\). By \(\mathbb {Z}/n\) denote the cyclic group of order \(n\) with discrete topology and by \(S^2\) the 2-sphere with standard topology. Let \(p\) and \(q\) be two distinct points of \(S^2\). Compute the homology groups of the \(n\)-link sausage space \(X\) which is the space that can be represented as the identification space \((S^2\times \mathbb {Z}/n)/{\sim }\) with the equivalence relation induced by \((p,i)\sim (q,i+1)\).
Diagram description: the space \(X\) is drawn as a cyclic chain of lobe-shaped copies of \(S^2\). Each copy is attached to the next one around the cycle, with the \(p\)-point of one copy identified with the \(q\)-point of the next.