Degrees of maps in algebraic topology

Question

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?

Answer 1

Welcome user48617,

Are you familiar with the Künneth-Theorem and cellular homology?

In order to construct maps that start in $\mathbb CP^1 \times \mathbb CP^2$ or $S^2 \times S^2$ it suffices to construct maps that start in one of the factors using the cell complexes of the respective spaces.

By Künneth Theorem, you get an induced homomorphism of short exact sequences but here the $\mathrm{TOR}$-term vanishes, i.e. the degree of the map in question is in fact the product of the degrees of the simpler maps.

Answer 2

For maps involving projective spaces, an elementary technique is to try to find a surjection involving homogeneous polynomials-you have coordinates in this case, so you should use them. Given polynomial maps $p:\mathbb{C} P^1\to \mathbb{C} P^3$ and $q:\mathbb{C} P^2\to \mathbb{C} P^3$ I can construct $pq$ by multiplying coordinatewise: adding wouldn't preserve well-definedness, but multiplying does, as in $([z:w],[a:b:c])\mapsto [za^2:wb^2:zc^2:w(a^2+2b^2+3c^2)].$ A similar technique might do the $S^2\times S^2\to \mathbb{C} P^2$ case, since that's also a map out of a product of projective spaces.

Answer 3

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).