# immersion of punctured torus in plane

Let $S^1 \times S^1$ be the $2$-torus. If a point $a=(p,q)$ of the torus is removed, i.e., it is punctured at one point then how can I show that it can be immersed in the plane, i.e., in $\Bbb{R}^2$?

I have the idea of decomposing the punctured torus into two intersecting cylinders and then immersing each of them to $\Bbb{R}^2$, but the problem is by this way even if we make that the intersecting region mapped to same region, we cannot define an immersion on the whole the two immersions may not agree pointwise on the intersection. I have got stuck here.

Thanks in advance for any kind of help.

• @tandra b) Follows from exercise 4 (next exercise): Consider the product of $n$ circles (''$1$-spheres'') and remove a point $p$. Once you have immersed $S^1\times S^1\{p\}$ in $\mathbb R^2$, what you get is a product of $n-2$ $1$-spheres times $R^2$. According to the answer provided here that embeds into $\mathbb R^{n-2+1}\times \mathbb R^= \mathbb R^n$. Oct 24, 2019 at 15:43
• I think there's trouble with factors: $S^1\times S^1\times S^1\setminus\{(a,b,c)\}$ and $S^1\times\big(S^1\times S^1\setminus\{(b,c)\}\big)$. Oct 25, 2019 at 16:42