Understanding the question of Hatcher 1.1.6

Question

In Hatcher 1.1.6

We can regard $\pi_1(X,x_0)$ as the set of basepoint-preserving homotopy classes of maps $(S_1,s_0) \to (X,x_0)$. Let $[S_1,X]$ be the set of homotopy classes of maps $S_1 \to X$, with no conditions on basepoints. Thus there is a natural map $Φ: \pi_1(X,x_0) \to [S_1,X]$ obtained by ignoring basepoints.

I don't quite understand what is mean by "with no condition on base points"(What is mean by "base point" of $S_1$?) and "ignoring basepoints". And whether $[S_1,X]$ is defined to be a set of continuous map or all the maps. What is a formal description of "ignoring basepoints"?

If some one could give an example that given a map in $\pi_1(X,x_0)$, how to get maps in $[S_1,X]$ in the way the author describes. I would very appreciate.

Answer 1

Map here does mean continuous function. Here $[S^1,X]$ is the set of equivalence classes of continuous maps from $S^1$ to $X$, where two maps are equivalent if they are homotopic. On the other hand $\pi_1(X,x_0)$ is the set of equivalence classes of "basepoint-preserving" continuous maps $S^1\to X$ where two maps are equivalent by a "basepoint-preserving" homotopy. So this is a smaller class of maps factored by a finer equivalence, so we get a map from $\pi_1(X,x_0)$ to $[S^1,X]$.

In the text Hatcher defines $\pi_1(X,x_0)$ in a different way, as the set of continuous maps from $I$ to $X$ with $f(0)=f(1)=x_0$ factored out by a suitable equivalence relation. But a continuous map from $I$ with $f(0)=f(1)$ is the same as a map from $S$ where $S$ is the quotient space of $I$ got by identifying $0$ and $1$. But this $S$ is homeomorphic to $S^1$. If we let $s_0$ be the point on $S=S^1$ to which $0$ and $1\in I$ are identified, then continuous maps from from $I$ to $X$ with $f(0)=f(1)=x_0$ are the same as continuous maps from $S^1$ to $X$ with $f(s_0)=x_0$, that is "basepoint-preserving" maps. Also homotopies need to be "basepoint-preserving" too...

Answer 2

An element $\gamma\in\pi_1(X,x_0)$ is an equivalence class of functions. Take one function from $\gamma$. It is (or can, at least, be seen as) a function from $S^1$ to $X$ where a specific point of $S^1$ is sent to $x_0$ (i.e. base point preserving).

However, at the same time it is also just a function from $S^1$ to $X$ (we just "forget" that the function has this base point preserving property). Therefore it is part of some equivalence class in $[S^1,X]$. The point of Hatcher is that we get the same equivalence class of $[S^1,X]$ no matter which representative of $\gamma$ we choose.