I have a past qual question here: characterize the space $[(\mathbb{RP}^2,x),(S^1 \times S^3,y)]$ of homotopy classes of maps from $(\mathbb{RP}^2,x)$ to $(S^1 \times S^3,y)$, where here $x \in \mathbb{RP}^2$ and $y \in S^1 \times S^3$ are base points.
In cases where we are considering homotopy classes of maps into $S^1$, I've used the relation $[X,S^1] = H^1(X;\mathbb{Z})$ when $X$ is homotopy equivalent to a CW complex. Is there a similar relation in this case? If so, does it have to do with changing coefficients?
Thanks in advance for your help!
By cellular approximation, the image of any map $\Bbb RP^2 \to S^1 \times S^3$ lies in a copy of $S^1$ inside $S^1 \times S^3$. Now use covering spaces to show that any map $\Bbb RP^2 \to S^1$ is nullhomotopic.