Inclusion of $n$-skeleton induces surjection of cohomology rings

Question

Suppose $X$ is a CW complex and $X_n$ its $n$-skeleton, the cohomology ring $H^*(X)$ of $X$ is defined to be the direct sum of its cohomology groups with multiplication as cup product. Does the inclusion $i:X_n\rightarrowtail X$ induce surjection of the cohomology rings $i^*:H^*(X)\to H^*(X_n)$ ? I can see that it is isomorphism for $i^k:H^k(X)\to H^k(X_n)$ with $k<n$ and $H^k(X_n)=0$ for $k>n$. But how can I determine whether $i^n:H^n(X)\to H^n(X_n)$ is surjective?

Answer 1

The inclusion $S^1 \rightarrow S^\infty$ cannot induce a surjection since $S^\infty$ is contractible.