Forgetting basepoints map is neither injective nor surjective.

Question

Let $X$ and $Y$ be topological spaces with basepoints $x_0 \in X$ and $y_0 \in Y$. Forgetting basepoints defines a map $$\Phi:[(X,x_0),(Y,y_0)]_∗ →[X,Y].$$

We write $[X, Y ]$ for the set of homotopy classes of continuous maps from $X$ to $Y$.

And we write $[(X, x_0), (Y, y_0)]_∗$ for the set of basepoint preserving homotopy classes of basepoint preserving continuous maps from $X$ to $Y$.

I want to show that in general $\Phi$ is neither injective nor surjective.

For a counterexample for surjectivity, we can consider $X= \{x_0\}$ and $Y=\{y_0,y_1\}$.

But I do not have and idea for a counterexample of injectivity.

Answer 1

In general, the set of homotopy classes of maps $S^1 \to X$ corresponds to the conjugacy classes of $\pi_1(X, x_0) = [(S^1, 1), (X, x_0)]_*$. So any two elements of $\pi_1(X, x_0)$ which are conjugate but not equal will provide an example where injectivity fails. Note, for such elements to exist, $\pi_1(X, x_0)$ must be non-abelian.

It's not a great picture, but the blue and red loops below are freely homotopic (that is, they define the same element of $[S^1, X]$), but not homotopic relative to the basepoint $x_0$ (that is, they define different elements of $\pi_1(X, x_0)$). In $\pi_1(X, x_0)$, they are conjugate via the green loop.