Higher homotopy of wedges of CW-complexes

Question

Is $\pi_{k}(\mathbb{S}^n\vee X)$ is isomorphic with $\pi_{k}(X)$, where $k < n$ and $X$ is a finite CW-complex. If yes, could you please give me a reference for such a staff?

Answer 1

Yes, and a very direct way to see this is as follows.

The sphere $S^n$ has a CW structure with one $0$-cell and one $n$-cell. Since $X$ is a CW-complex it then follows that so is the wedge $S^n\vee X$, and moreover that its $k$-cells for $k<n$ are exactly those of $X$.

Using cellular approximation we can assume that any class $\alpha\in\pi_k(S^n\vee X)$ is represented by a cellular map $\alpha:S^k\rightarrow S^n\vee X$. Write $i:X\hookrightarrow S^n\vee X$ for the canonical inclusion. Then, from the previous comments, the cellular map factors up to $\alpha:S^k\xrightarrow{\tilde\alpha}X\xrightarrow{i} S^n\vee X$. Thus the induced map $i_*:\pi_kX\rightarrow \pi_k(S^n\vee X)$ is a surjection for $k\leq n$.

Finally observe that if $q:S^n\vee X\rightarrow X$ is the pinch map and $\beta\in\pi_kX$ and $F:i_*B\simeq\ast$ is a null-homotopy of the composite $S^k\xrightarrow{\beta} X\xrightarrow{i} S^n\vee X$, then $q\circ F:S^k\times I\rightarrow X$ is a homotopy $(q\circ i)_*\beta=(id_X)_*\beta=\beta\simeq q_*\ast=\ast$. That is $q\circ F$ is a null-homotopy of $\beta$. Thus $i_*:\pi_kX\rightarrow \pi_k(S^n\vee X)$ is injective (this is true for all $k\geq 0$).

We conclude that $\pi_k(S^n\vee X)\cong\pi_kX$ for $k< n$.