Homotopy classes of maps out of a wedge of circles

Question

Let $W_n$ be the wedge on $n$ circle, i.e.,
$$
W_n=\bigvee_{k=1}^nS^1
$$
Let $X$ be a path connected space, then how do I calculate the homotopy classes of mas from $W_n$ to $X$? The based maps are easy as the wedge is the coproduct in the category of pointed spaces, thus
$$
\mathrm{hTop}_*\left[\bigvee_{k=1}^nS^1,X\right]=\prod_{k=1}^n\mathrm{hTop}_*\left[S^1,X\right]=\prod_{k=1}^n\pi_1\left(X\right)
$$
Is something similat true for non-pointed maps?

Answer 1

When $X, Y$ are path-connected, the set $[X, Y]$ of homotopy classes of maps relates to the set $\langle X, Y \rangle$ of pointed homotopy classes of pointed maps by $$[X, Y] = \langle X, Y\rangle /\pi_1(Y),$$ where the action of $\pi_1(Y)$ is given by finding a map $\gamma \cdot f: X \to Y$ with an unbased homotopy $F_t: f \to \gamma \cdot f$ (that is, $F_0 = f$ and $F_1 = \gamma \cdot f$) so that $F_t(x_0) = \gamma(t)$. See Hatcher 4A.1.

In your case you already know that $\langle W_n, Y\rangle = \pi_1(Y)^n$. The action of $\pi_1(Y)$ on this set is the diagonal action where we have the conjugation action in each component; so $[W_n, Y]$ is the set of ordered $n$-tuples $(\gamma_1, \cdots, \gamma_n)$ considered up to conjugacy --- so $$(\gamma_1, \cdots, \gamma_n) \sim (\eta \gamma_1 \eta^{-1}, \cdots, \eta \gamma_n \eta^{-1}).$$