Degree $1$ map from torus to sphere

Question

I'm trying to find a smooth degree 1 map from the torus $T^2 = S^1 \times S^1$ to the $2$-sphere $S^2$.

My first thought was to use the two coordinates $(\theta_1,\theta_2)$ to map onto the usual spherical polar coordinates on the sphere - however I can't quite get that to work because it seems that one of these coordinates would cover the sphere twice and so give a degree $2$ map?

Alternatively I thought about using the winding number. So in general if we embed circles into $\mathbb{R}^3$ by the maps $f_1,f_2: S^1 \to \mathbb{R}^3$ we can then define a map $F:T^2 \to S^2$ by:

$$F(\theta_1,\theta_2) = \frac{f_1(\theta_1) - f_2(\theta_2)}{\| f_1(\theta_1) - f_2(\theta_2) \|}$$

However I'm now struggling to make suitable choices for $f_1$ and $f_2$, my guess is that I'd want to make the circles cross over each other once but I'm struggling to visualise this and write down a map so I would appreciate any help.

Answer 1

Here's a way to construct a degree one map.

Consider the standard CW complex for the torus consisting of a $0$-cell, two $1$-cells, and one $2$-cell. If we forget about the $2$-cell, what remains is $S^1\vee S^1$. That is $T^2 = (S^1\vee S^1)\cup_{\varphi} e^2$ where $e^2$ is the $2$-cell and $\varphi$ is the attaching map $\varphi : \partial e^2 \to S^1\vee S^1$.

Now consider the space $T^2/(S^1\vee S^1)$ where all the points of $S^1\vee S^1$ are identified with one another. This quotient space has the CW structure $e^0\cup_{\tilde{\varphi}} e^2$ where $e^0$ is a $0$-cell (i.e. a point) and $\varphi : \partial e^2 \to e^0$. We see that the quotient is $S^2$ and so we can view the quotient map as a map $f : T^2 \to S^2$.

As $f|_{T^2\setminus(S^1\vee S^1)} : T^2\setminus(S^1\vee S^1) \to S^2$ is a homeomorphism (it identifies the $2$-cell of $T^2$ with the $2$-cell of $S^2$), we see that $f$ is a degree one map $T^2 \to S^2$.

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The above argument can be used to show that any connected closed smooth $n$-manifold $M$ admits a degree one map to $S^n$. Using Morse theory, one can show that $M$ admits a CW complex structure with only one $n$-cell, i.e. $M = M^{(n-1)}\cup_{\varphi}e^n$ where $M^{(n-1)}$ is the $(n-1)$-skeleton of $M$. Then $M \to M/M^{(n-1)} \cong S^n$ is a degree one map.

We needed to assume $M$ was smooth above in order to use Morse theory. However, for $n \neq 4$, every closed connected $n$-manifold has such a CW complex structure with one $n$-cell, even if it isn't smoothable, so all such manifolds admit a degree one map to $S^n$. If $n = 4$, it is not known whether such a CW complex structure exists (in fact, it is not even known if every closed four-manifold has a CW complex structure). See this MathOverflow question for more details.

Answer 2

Let $(U, \varphi)$ be a chart on $T^2$, i.e. $U$ is an open subset of $T^2$ and $\varphi : U \to \mathbb{R}^2$ is a homeomorphism. Let $V$ be the open subset of $U$ such that $\varphi|_V : V \to B(0, 1)$ is a homeomorphism, i.e. $V = \varphi^{-1}(B(0, 1))$. Then $\varphi(\overline{V}) = \overline{\varphi(V)} = \overline{B(0, 1)}$ and $\varphi(\partial V) = \partial\varphi(V) = \partial B(0, 1) = S^1$.

Note that the quotient $\overline{B(0, 1)}/S^1$ is homeomorphic to $S^2$; let $\psi : \overline{B(0, 1)}/S^1 \to S^2$ be a homeomorphism. The composite $\psi\circ\varphi|_{\overline{V}} : \overline{V} \to S^2$ maps $\partial V$ to a single point, call it $p$, and $(\psi\circ\varphi|_{\overline{V}})|_V = \psi\circ\varphi|_V$ is a homeomorphism from $V$ to $S^2\setminus\{p\}$.

Now define $f : T^2 \to S^2$ by

$$f(x) = \begin{cases} \psi(\varphi(x)) & x \in \overline{V}\\ p & x \not\in \overline{V}. \end{cases}$$

Then $f$ is a continuous map. Furthermore, it has degree one .

More generally, we can use the same technique to construct a degree one map from any closed, connected, orientable $n$-manifold to $S^n$.

Answer 3

Your idea about using linking numbers is certainly on the right track. Consider the Hopf link. If we think of the circles as being embedded in two perpendicular planes then one can understand the map \begin{align} S^1 \times S^1 &\to S^2 \\ (x,y) &\mapsto \frac{x-y}{\|x - y\|} \end{align} by drawing the images of a circle for each $x$ fixed. If you think about this for a little while you realize that this can be thought of as placing the sphere in the interior of the torus and then flattening the torus onto the sphere. At this point one can see that the map generically has 3 or 1 preimages and wherever it has 1 preimage, it is orientation preserving so the map has degree 1.

Answer 4

Take lat/lon coordinates on the sphere: latitude $\phi$ from 0 to $\pi$, longitude $\theta$ from $o0$ to $2\pi$.

Take polar coordinates $\alpha, \beta$ on the torus, both running from $0$ to $2\pi$.

Let $$ a = \frac{\alpha - \pi}{\pi} \\ b = \frac{\beta - \pi}{\pi} $$ be coordinates with the range $-1$ to $1$ on the torus.

Now consider $$ f(a, b) = \begin{cases} \sqrt{1 + \frac{\min(|a|, |b|)^2}{\max(|a|, |b|)^2}} & \text{either $a$ or $b$ nonzero}\\ 0 & \text{a = b = 0} \end{cases}. $$ This is the length of a ray through $(a, b)$ hitting the edge of the square $S = \{(x, y) \mid -1 \le x,y \le 1 \}$.

Define $$ g(a, b) = \frac{1}{f(a, b)} (a, b) $$ This maps the square $S$ to the unit disk $D$ (essentially by "radial stretching").

Finally, define $$ h: D \to S^2 : (u, v) \mapsto (\pi\sqrt{u^2+v^2}, atan2(u, v) ) $$ where points of $S^2$ are represented in $(\phi, \theta)$ coordinates.

Now the map $h \circ g$ takes $(a, b)$ coordinates on the torus to $(\phi, \theta)$ coords on the sphere, and you've got your map.

Shorthand: take the square $S$, which is the domain for $(a, b)$ coordinates on the torus, and send its whole boundary to the south pole, and its center to the north pole.

Answer 5

Toroidal coordinates are generally reconstructed from Cartesian, cylindrical, or spherical coordinates. The problem is the skewness of the toroidal basis vectors. Without redefining the isomorphism, you can just select regions where the sphere to torus mapping is defined. Why do you need to do this? Toroidal eigenfunctions and harmonics exist. Maybe you want to improve these. A nice idea is to look at spherical modes and then map them to toroidal modes. This could be useful in toroidal structures like transformers with fluid, tokamaks, and pipes with toroidal flow. I can show you my calculations, but you might find research into the Gamma function and toroidal coordinates interesting. The current problem is the modulation of the cosine and hyperbolic cosecant. It seems mathworld cannot get the sign of the cosine correct. You have stumbled onto an unknown problem. Follow this link for the quick and dirty mapping. https://en.m.wikipedia.org/wiki/Toroidal_coordinates Good luck.