# Proving the degree of the antipodal circle function.

I was asked to determine the degree of the antipodal circle function $$f(\theta)=\theta + \pi$$. I believe the degree of the function is $$1$$, through pictures and illustration, but I will like to prove this using homotopy.

So given our degree $$1$$ map $$C_{-1}(\theta):S^1 \rightarrow S^1$$ is given by $$f(\cos(\theta)+i\sin(\theta))=\cos(\theta)+i\sin(\theta)$$.

Could I use as a homotopy $$H(x,t)=(\cos(\theta)+i\sin(\theta))e^{t\pi i}$$. If so how will prove it is continuous.

Yes, you can. However, you can simplify your formula by writing $$H(z,t) = z\cdot e^{\pi i t} .$$ Note that $$S^1 = \{ z \in \mathbb C \mid \lvert z \rvert = 1\}$$. The map $$H$$ is continuous because complex multiplication is continuous and the exponential function is continuous.
If you want you can also write $$S^1 = \{ (x,y) \in \mathbb R^2 \mid x^2 + y^2 = 1\}$$ and $$H(x,y,t) = (x\cos(\pi t) - y \sin(\pi t), x \sin(\pi t) + y \cos(\pi t)) .$$
• Thank you but this does not prove our function is continuous. The space $S^1\times[0,1]$ im not sure if it a metric space. Hence, what are you usig to prove that our function is continuous. I beloeve we need an intermediate function. May 23, 2020 at 18:11
• It does prove it, but it takes for granted a few well-known facts. I am not going to give proofs. 1) The function $[0,1] \to \mathbb C, t \mapsto \pi i t$, is continuous. This is trivial, multiplication with the constant $c = \pi i$ is continuous. 2) The function $\mathbb C \to \mathbb C, w \to e^{iw}$, is continuous. Complex multiplication $\mathbb C \times \mathbb C \to \mathbb C, (z_1,z_2) \mapsto z_1 \cdot z_2$, is continuous. May 23, 2020 at 23:22
• Wouln't we also need that that if the function $\mathbb{C} \rightarrow \mathbb{C}$ and $[0,1] \rightarrrow \mathbb{C}$ is continuous. Then $\mathbb{C} \times \rightarrow \mathbb{C} \times \mathbb{C}$ is continuous. I know a lot steps, I just need my proofs alwas to be true. May 24, 2020 at 7:10
• @benhuni Yes, you are right. We need the fact that "product maps" (here $\mathbb C \times [0,1] \to \mathbb C \times \mathbb C$) are continuous. This is true for all pairs of maps between metric (or more generally topological) spaces. By the way, you can also an $\epsilon$-$\delta$-argument to show directly the $H(x,y,t)$ is continuous. May 24, 2020 at 8:24