How to compute the cohomology $H^{*}(S^{n}, S^{k})$?

Question

Let $S^{k} \subset S^{n}$ be the $k$-sphere included in the $n$-sphere as the unit sphere in the first $k +1$ coordinates sitting in the unit sphere in $(n+1)$-space. Compute $H^{*}(S^{n}, S^{k})$.

Can this question be solved with Mayer-Vietoris sequence? Any hints?

Thank you in advance.

Answer 1

(Thorben Kastenholz advices very nice solution using long exact sequence; here is homotopic one)

the factor $S^n/S^k$ is homotopy equivalent to $S^n\cup CS^k$, and it is homotopy equivalent to $S^n\vee S^{k+1}$. (here $C$ is a cone)

so we have $H^i(S^n,S^k)=\mathbb Z$ for $i=k+1,n$, and $0$ otherwise. in case $n=k+1$ we have $H^n=\mathbb Z^2$.