Homeomorphism from $S^2 \times S^2$ to a subset of $\mathbb{R}^5$

Question

I am beginner in topology and I have this exercise in the book I'm reading:

Describe a homeomorphism from $S^2 \times S^2$ to a subset of $\mathbb{R}^5$. (Hint: identify $S^2 \times S^2$ with a subset of $S^5$)

I had the idea of identifying $S^2\times S^2$ with $$E:=\{(x,y,z,t,v,w)\in \mathbb{R}^6\mid x^2+y^2+z^2=\frac 12 = t^2+v^2+w^2\}\subsetneq S^5$$ with the homeomorphism $\sigma:S^2 \times S^2\to E$ : $(\,\,(x,y,z),(t,v,w)\,\,)\mapsto \,\frac{\sqrt{2}}{ 2}(x,y,z,t,v,w)$.

Now, as $S^5\subseteq \mathbb R^6$ and the exercise asks for $\mathbb R^5$, I thought we could compose $\sigma$ with a stereographic projection: $S^5\setminus\{P\}\to \mathbb R^5$ where $P$ is any point in $S^5\setminus E$, for example $P=(1,0,0,0,0,0)$.

Is this correct ?

Thanks in advance !

Answer 1

Since the normal bundle of $S^2$ in $\mathbb R^5$ is trivial, the boundary of its tubular neighborhood in $\mathbb R^5$ is exactly $S^2\times S^2$.

Answer 2

Here is an explicit geometric version based on the embedding of $S^1\times S^1$ as torus in $\Bbb R^3$. Embed $S^2$ into $\Bbb R^3\subset\Bbb R^5$. Choose

$$e_1=(0,0,0,1,0),\quad e_2=(0,0,0,0,1).$$

Then we can define the embedding

$$\iota:S^2\times S^2\to\Bbb R^5,\quad (x,[y_1\;y_2\;y_3]^\top)\mapsto 3x+(y_1e_1+y_2e_2+y_3x).$$

Here $[y_1\; y_2\; y_3]^\top$ is a point on the second $S^2$ written in coordinates, i.e. $y_1^2+y_2^2+y_3^2=1$. This naturally generalizes to embeddings of $S^n\times S^n$ into $\Bbb R^{2n+1}$