Let $\phi_{n,k}$ denote $\Sigma^{n-1} \phi_k : S^n \longrightarrow S^n.$ Compute $H_*(C(\phi_{n,k})),$ where $C(\phi)$ is the mapping cone of $\phi.$
Here $\Sigma^{n-1}$ denotes the $(n-1)$-fold reduced suspension and $\phi_k$ denotes multiplication by $k.$
How do I compute it? It turns out to me that the problem can be done using Mayer-Vietoris long exact sequence if it is known what's the effect of $\Sigma^{n-1} \phi_k$ in the level of homology. But I am unable to figure out the induced homomorphism. Any help in this regard would be much appreciated.
Thanks in advance.