Constructing maps of degree $k$

Question

One of the common constructions one finds when first learning about the (topological) degree of a map is the construction of maps $f_k:S^n\rightarrow S^n$ of degree $k\in\mathbb{Z}$ (i.e. $f_k(z)=z^k$).

I am curious: are there general constructions available for degree $k$ maps between other spaces? For example, is there an explicit construction of, say (to slightly generalize the above construction), degree $k$ maps from a compact Riemann surface to another compact Riemann surface?

I think that just getting rid of the extra loops will yield a degree 1 map from a (compact) Riemann surface of genus $g$ to any (compact) Riemann surface of genus less than $g$… but I'm not sure how to get to arbitrary degree… any suggestions?

Answer 1

There are no general constructions because for a closed, connected, oriented manifold $X$, there need not be a map $f : X \to X$ of degree $k$ unless $k = 0$ (e.g. constant maps) or $k = 1$ (e.g. the identity map). An example of a manifold which only admits self-maps of degree $0$ and $1$ is $(\Sigma_2\times\Sigma_2)\#\overline{\mathbb{CP}^2}$; this follows from the properties of the Gromov norm and the fact that a manifold with non-zero signature cannot have a self-map of degree $-1$.

Using the fact that $\deg(f\circ g) = \deg f\cdot\deg g$, one thing you can say is that the set of degrees of self-maps of $X$ is closed under multiplication.

As for degrees of maps between Riemann surfaces, we have the following (see this answer):

• if $g < g'$, every map $\Sigma_g \to \Sigma_{g'}$ has degree zero;
• if $g \geq g'$ and $g' = 0$ or $1$, there is a map $\Sigma_g \to \Sigma_{g'}$ of every degree;
• if $g \geq g' \geq 2$, there is a map $\Sigma_g \to \Sigma_{g'}$ of degree $k$ if and only if $k$ satisfies $|k| \leq \left\lfloor\dfrac{g-1}{g'-1}\right\rfloor$.

Answer 2

You can find a lot of degree $n$ maps from a Riemann surface to the Riemann sphere.

For example, if the Riemann surface $S$ arises as a projective algebraic curve in the complex projective plane, then linear projection from a point outside the Riemann surface gives a (ramified) cover of the sphere by $S$.

(Linear projection from $p = [1:0: \ldots :0]$ is the map that sends $[a_0 : \ldots : a_n]$ with some $a_i \not = 0$ for $i > 0$, to $[a_1 : \ldots : a_n]$, in homogeneous coordinates. This is defined everywhere except for p, and geometrically is the map that sends a point $q$ to the line between $p$ and $q$. Locally this looks like coordinate projections maps. Note that there is a projective line $\mathbb{C}P^1$ of lines through $p$, and this is the target space.)

To summarize, we fix a point $p$, and send each point $q \in S \subset \mathbb{C}P^2$ to the line through $q$ and $p$.

If the algebraic curve is defined by an equation of degree d, then Bezout's theorem says that a generic line will intersect a curve of degree $d$ in $d$ points. This applies also to a generic line through $p$, provided that $p$ was not on the curve. (The map from $S$ to $\mathbb{C}P^1$ can only be ramified at finitely many points, since it isn't constant if we assume $S$ is not a line, so most fibers will be unramified, meaning that the corresponding line will intersect $S$ transversally, meaning that it will intersect $S$ at $d$ points. I'm using that if a line $L$ intersect an algebraic curve $S$ of degree $d$ transversally, then it will intersect in $d$ points - this is an easy consequence of a more precise statement of Bezout's theorem. I think you can also see this by studying the intersection of the dual curve of $S$, which is an irreducible curve which is not a line, with the dual curve of $p$, which is a line - and so their intersection can only be finitely many points.)

Therefore a generic fiber of the induced map $S \to \mathbb{C}P^1$ has $d$ points in the fiber, so it is of degree $d$.

The topological degree of a map between two compact Riemann surfaces is the number of points in a generic fiber / the degree of the field extension between their fields of meromorphic functions / the induced map on top homology using the natural orientation of the surfaces coming from the complex structure.

There are limitations on the genus of the surfaces that can arise this way, because of the genus degree formula. You can (famously) produce elliptic curves this way using the Weiestrass equation, as well as curves of arbitrarily large genus.

However, this still gives a lot of explicit examples. You can also try to play similar games with higher dimensional varieties in complex projective space. There is an incredibly detailed discussion here: How do different definitions of "degree" coincide?