I've got a question about the solution of the following exercise:
Let $X$ be a connected CW-complex and $X^n$ its $n$-skeleton (i.e. the subcomplex of all cells of dimension $n$ or less). Denote by $i\colon X^n \to X$ the inclusion and let $x_0 \in X^n$ be any base point. Show that $i_*\colon \pi_k(X^n, x_0) \to \pi_k(X, x_0)$ is injective for $k \le n-1$.
The solution says right at the beginning:
Suppose that $k \le n-1$ and $[f] \in \pi_k(X^n, x_0)$ lies in the kernel of $i_*$. First of all we may assume $f$ is cellular. There is a base point preserving homotopy $H:S^k\times [0,1]\to X$ from $f$ to the constant map.
My questions:
1) Why is there a base point preserving homotopy from $f$ to the constant map? I struggle to see why $f$ is nullhomotopic.
2) We've chosen $[f] \in \pi_k(X^n, x_0)$, thus $f$ is a map given by $$f\colon S^k \to X^n.$$ The homotopy $H$ however seems to start at $h_0 = f\colon S^k \to X$ which would represent an element of $\pi_k(X,x_0)$ rather than $\pi_k(X^n, x_0)$. Or is this still valid since $X^n \subseteq X$?
Appreciating any help!
Thank you very much.
I think the issue is that the solution is worded a bit strangely.
We are concerned with the injectivity of the map $i_*\colon \pi_k(X^n) \to \pi_k(X)$, so if we assume $[f]\in \pi_k(X^n)$ is in the kernel of $i_*$ then by definition there is a basepoint-preserving homotopy from $i\circ f\colon S^k \to X$ to the constant map. (We could consider $f$ and $i\circ f$ to be "the same" since $i$ is an inclusion, but they are technically different.)
The question now is whether or not $[f] = 0$ i.e. whether there is a basepoint preserving nullhomotopy of $f\colon S^k \to X^n$. But suppose $H\colon S^k \times [0,1] \to X$ is a basepoint-preserving homotopy from $i\circ f$ to the constant map. If we assume that $f$ is cellular then $H$ is already cellular on the subcomplex $K = S^k \times \{0\} \cup \{e_1\} \times I$, then by cellular approximation $H$ is homotopic to a cellular map $\tilde{H}\colon S^k\times [0,1] \to X$ which agrees with $H$ on $K$. But since $k \leq n-1$ and $dim(S^k\times [0,1]) = k + 1$ it follows by cellularity that the image of $\tilde{H}$ is actually in $X^n$, or in other words it is a basepoint-preserving nullhomotopy of $f$ in $X^n$ so $[f] = 0 \in \pi_k(X^n)$.