Immersing punctured torus

Question

I am looking for a proof (as elementary as possible) of the fact that punctured n-dimensional torus admits an immersion to $\mathbb{R}^n$. The 2-dimensional case seems to be evident, but I haven't got the idea how to give an explicit construction in the n-dim case.

Answer 1

The general case is very much like the 2-dimensional case, it just takes time to process the picture, to see how you could do the same constructions in the higher-dimensional case.

A punctured $S^1 \times S^1$ looks like a wedge of two circles, but fattened up a little bit. Precisely, around each circle you have an annulus neighbourhood. To immerse the punctured torus into $\mathbb R^2$ what you do is you embed the first annulus, and then embed the 2nd annulus so that it overlaps the first in the same way that the two annuli live in the torus itself. In doing this you create an extra overlap, but that's fine as we're only looking for an immersion.

A punctured $S^1 \times S^1 \times S^1$ has the same kind of decomposition. It looks like the union of three "annuli" , precisely, $[0,1] \times S^1 \times S^1$, $S^1 \times [0,1] \times S^1$ and $S^1 \times S^1 \times [0,1]$, where here $[0,1]$ is shorthand for a small interval in $S^1$. Each of these spaces you can embed in $\mathbb R^3$ as tubular neigbhourhoods of embedded tori. You just have to make the embeddings overlap in the same way they overlap in $S^1 \times S^1 \times S^1$ -- and that is to make $[0,1] \times S^1 \times S^1 \cap S^1 \times [0,1] \times S^1 = [0,1]^2 \times S^1$, i.e they intersect along one of the coordinate circles. So the idea is to draw a picture of an embedded torus, then along each of the two coordinate (longitude/latitude) axis, draw the boundary of a tubular neighbourhood of that axis. Suitably interpreted, you can think of this picture as the image of the coordinate tori under your immersion.

The general picture goes like that.