Degrees of maps in algebraic topology Please can I have some tips on how to construct maps between topological spaces of a given degree? For example, how would you go about building a map of degree $3$ from $\mathbb{CP}^1\times\mathbb{CP}^2 \to \mathbb{CP}^3$? Or a map from $S^2\times S^2 \to \mathbb{CP}^2$ of even degree? I don't know where to start. Are there any particular techniques that are useful?

In general it is hard to write down maps of a given degree, or even to determine whether such maps exist.
I don't know if there is a map $\mathbb{CP}^1\times\mathbb{CP}^2 \to \mathbb{CP}^3$ of degree three, but there are maps $S^2\times S^2 \to \mathbb{CP}^2$ of even degree. In fact, every map $S^2\times S^2 \to \mathbb{CP}^2$ has even degree.
To see this, recall that $H^*(S^2\times S^2; \mathbb{Z}) \cong \mathbb{Z}[\alpha, \beta]/(\alpha^2, \beta^2)$ and $H^*(\mathbb{CP}^2; \mathbb{Z}) \cong \mathbb{Z}[\omega]/(\omega^3)$. Consider a map $f : S^2\times S^2 \to \mathbb{CP}^2$. Note that $f^*(\omega^2) = (f^*\omega)^2$ and since $f^*\omega \in H^2(S^2\times S^2; \mathbb{Z}) \cong \mathbb{Z}\alpha\oplus\mathbb{Z}\beta$, $f^*\omega = x\alpha + y\beta$ for some $x, y \in \mathbb{Z}$. Therefore
$$f^*(\omega^2) = (f^*\omega)^2 = (x\alpha + y\beta)^2 = 2xy\alpha\beta.$$
As $\alpha\beta$ is a generator of $H^4(S^2\times S^2; \mathbb{Z})$, $\deg f = \pm 2xy$ which is even (the sign depends on the orientations).
