I want to prove the following statement:
Let $(X, x_0)$ be a pointed space, and let $X' = X\cup_{\alpha} e^{n+1}$ be obtained from $X$ by adjoining an $(n + 1)$-cell. Then the inclusion $i : X\rightarrow X'$ induces a map $\pi_k(X, x_0)\rightarrow\pi_k(X', x_0)$ which is an isomorphism for $k < n$ and surjective for $k = n$.
We know that attaching a $n+1$-cell kills homotopies. Can this be used to prove the result? Or do we have to argue elementary? Can someone help?
Thank you very much.
This is a standard result which you will find in many books. In Topology and Groupoids the following is proved.
7.6.1 The following statements are true for $n \geqslant 1$:
$\alpha(n)$: Any map $S^r \to S^n$ with $r < n$ is inessential.
$\beta(n)$: Any map $S^r \to S^n$ with $r < n$ extends over $E^{r+1}$.
$\gamma(n)$: Let $B$ be path connected and let $Q$ be obtained from $B$ b y attaching a finite number of $n$ cells to $B$. Then any map $(E^r, S^{r-1}) \to (Q,B)$ with $r < n$ is deformable into $B$.
The proof is by induction by means of the implications $$ \gamma(n) \implies \alpha(n) \iff \beta(n) \implies \gamma(n+1). $$
The harder part is the last implication, which uses a subdivision of the $r$-cube.