Calculate the Euler-Poincaré characteristic of followin surfaces.

Question

Calculate the Euler-Poincaré characteristic of:

1. An ellipsoid.
2. The surfase $S=\left\{ \left(x,y,z\right)\in\mathbb{R}^{3}:x^{2}+y^{10}+z^{6}=1\right\} $.

Note: Not how to do this problem, I not really have clear concepts.

Answer 1

Before the demonstrations, remember a proposition:

Proposition $\bigstar$: Let $S\subset\mathbb{R}^{3}$ an compact connected surface; then one of the values $2,0,-2,\ldots,-2n,\ldots$ is assumed by the Euler-Poincaré characteristic $\mathcal{X}\left(S\right)$. Furthermore, if $S'\subset\mathbb{R}^{3}$ is other compact surface and $\mathcal{X}\left(S\right)=\mathcal{X}\left(S'\right)$, then $S$ is homemorphic to $S'$.

With this in mind we have the following:

1. The procedure is to find a homeomorphism $\varphi$ between the unit sphere $S^{1}$ and the ellipsoid which we denote as $\mathcal{E}$, where $\mathcal{E}=\left\{ \left(x,y,s\right)\in\mathbb{R}^{3};\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1\right\} $ and where $a,$ $b$ y $c$ are constant nonzero, this idea is expressed graphically in the following figure:

   $\underrightarrow{\qquad\qquad}{\varphi}$

   This homeomorphims is: $$ \begin{array}{cccl} \varphi: & S^{1} & \rightarrow & \mathcal{E}\\ & \left(x,y,z\right) & \mapsto & \left(ax,by,cz\right). \end{array} $$ This homeomorphims is well-defined because $\varphi\left(x,y,z\right)\in S$ due to $\frac{\left(ax\right)^{2}}{a^{2}}+\frac{\left(by\right)^{2}}{b^{2}}+\frac{\left(cz\right)^{2}}{c^{2}}=x^{2}+y^{2}+z^{2}=1$. Additionally, it is clearly injective, we see that it is surjective, indeed, given $\left(\overline{x},\overline{y},\overline{z}\right)\in S$ consider the point $\left(\frac{\overline{x}}{a},\frac{\overline{y}}{b},\frac{\overline{z}}{c}\right)$ which satisfies $\varphi\left(\frac{\overline{x}}{a},\frac{\overline{y}}{b},\frac{\overline{z}}{c}\right)=\left(\overline{x},\overline{y},\overline{z}\right)$, note that $\left(\frac{\overline{x}}{a}\right)^{2}+\left(\frac{\overline{y}}{b}\right)^{2}+\left(\frac{\overline{z}}{c}\right)^{2}=\frac{\overline{x}^{2}}{a^{2}}+\frac{\overline{y}^{2}}{b^{2}}+\frac{\overline{z}^{2}}{c^{2}}=1$, then $\left(\frac{\overline{x}}{a},\frac{\overline{y}}{b},\frac{\overline{z}}{c}\right)\in S^{1}$, so that, $\varphi$ is surjective. The continuity of $\varphi$ is inferred because the components functions are continuous functions. Therefore, $\mathcal{E}$, is homeomorphic to the unit sphere, then $\mathcal{E}$ is a connected compact surface, but by Proposition $\bigstar$ we have that one of the values $2,0,-2,\ldots,-2n,\ldots$ is assumed by the Euler-Poincare characteristic $\mathcal{X}\left(\mathcal{E}\right)$, but if $\mathcal{X}\left(\mathcal{E}\right)$ assume a different value to $2$ then $\mathcal{E}$ would be homeomorphic to a type of torus, which it means that the unit sphere is homeomorphic to a type of torus, which may not be possible because it contradicts the fact that be simply connected is a topological invariant, so that we have $\mathcal{X}\left(\mathcal{E}\right)=2$.

2. The procedure is to find a homeomorphism $\varphi$ between the unit sphere $S^{1}$ and $S$, this idea is expressed graphically in the following figure:

   $\underrightarrow{\qquad\qquad}{\varphi}$

   This homeomorphims is: $$ \begin{array}{cccl} \varphi: & S^{1} & \rightarrow & S\\ & \left(x,y,z\right) & \mapsto & \left(x,y^{\frac{1}{5}},z^{\frac{1}{3}}\right). \end{array} $$ This homeomorphims is well-defined because $\varphi\left(x,y,z\right)\in S$ due to $x^{2}+\left(y^{\frac{1}{5}}\right)^{10}+\left(z^{\frac{1}{3}}\right)^{6}=x^{2}+y^{2}+z^{2}=1$. Additionally, it is clearly injective, we see that it is surjective, indeed, given $\left(\overline{x},\overline{y},\overline{z}\right)\in S$ consider the point $\left(\overline{x},\overline{y}^{5},\overline{z}^{3}\right)$ which satisfies $\varphi\left(\overline{x},\overline{y}^{5},\overline{z}^{3}\right)=\left(\overline{x},\overline{y},\overline{z}\right)$, note that $\overline{x}^{2}+\left(\overline{y}^{5}\right)^{2}+\left(\overline{z}^{3}\right)^{2}=\overline{x}^{2}+\overline{y}^{10}+\overline{z}^{6}=1$, then $\left(\overline{x},\overline{y}^{5},\overline{z}^{3}\right)\in S^{1}$, so that, $\varphi$ is surjective. The continuity of $\varphi$ is inferred because the components functions are continuous functions. Therefore, $S$ is homeomorphic to the unit sphere, then $S$ is a connected compact surface, but by Proposition $\bigstar$ we have that one of the values $2,0,-2,\ldots,-2n,\ldots$ is assumed by the Euler-Poincare characteristic $\mathcal{X}\left(S\right)$, but if $\mathcal{X}\left(S\right)$ assume a different value to $2$ then $S$ would be homeomorphic to a type of torus, which it means that the unit sphere is homeomorphic to a type of torus, which may not be possible because it contradicts the fact that be simply connected is a topological invariant, so that we have $\mathcal{X}\left(S\right)=2$.