Hatcher's 4.1.15: Not even understanding the question! Good people, I'm having problems even understanding some aspects of Hatcher's Ex. 4.1.15, which makes it rather difficult to actually solve it.
The question looks like this:

My questions are as follows:

*

*What does Hatcher mean when he writes "Show that every map $f: S^n \rightarrow S^n$ is homotopic to a multiple of the identity map"? (own emphasis.) Multiple in what sense? What is the "scalar" here?

*What does Hatcher mean in (b) when he writes "collapses the complement of a small ball about $q$ to the basepoint"? (own emphasis.) What is the basepoint of the map in this context?

Look forward to you sorting this out for me. No doubt this is fairly trivial.

Question 1: In earlier portions of the book, you have learned that $\pi_n(S^n)$ is an abelian group (assuming that $n \ge 2$). Hatcher is taking advantage of that fact in part (a), which allows him to use some shortcuts of terminology.
One can re-expand that shortcut by writing

... every map $f : S^n \to S^n$ is homotopic to a map that represents the same element in $\pi_n(S^n)$ as a multiple of the element represented by the identity map.

And, of course, in the context of an abelian group, a "multiple" of an element means an integer multiple; here one uses the fact that abelian groups are $\mathbb Z$-modules. For the case of a positive integer $n$ and the identity map $id$, this is written out explicitly in the comment of @КряжевАрсений:
$$n[id] = \underbrace{[id] + [id] +... + [id]}_{\text{$n$ times}}
$$
Question 2: In a context where one is working with a topological space and its homotopy group $\pi_n(X)$, such as part (b), the notation $\pi_n(X)$ is itself a shortcut for an expanded notation $\pi_n(X,b)$, where one implicitly assumes that a base point $b \in X$ has been chosen. So the "base point" referred to in part (b) is that implicitly chosen basepoint.
