Intuitive idea of immersions?

Question

So I understand the definition of immersions and submersions, as well as the motivation for defining such ideas. Not only are they important in understanding properties of mappings of tangent spaces, but they are also one of the few properties that remain invariant under homotopies. However, I am having trouble picturing these types of concepts. For example, how would one know intuitively when a surface can be immersed into the plane $\mathbb{R}^{2}$ without constructing an explicit map and taking the differential to see if it is injective or not?

For example, with the punctured tori, how does one see that the construction of it by taking the intersection of two cylinders along a square region is indeed an immersion into the plane? Or that the klein bottle can be immersed in 3-space?

Answer 1

You can help your intuition by saying: an immersion is locally an embedding. But this does not need to be globally. So what can happen is, that your immersed image contains self intersetcions. E.g. the Klein bottle can only be immersed in your mind, since it thinks in 3d. But it is obvious that an $n$-manifold can easier be immersed into some $n+k$-manifold for small $k$, than embedded.

I hope that this brief notice helps a bit.

Answer 2

Embeddings intuitively are easier to understand since we have several examples in low dimensions that can be immersed but cannot be embedded, like projective plane(cross-cap, Boy's surface) and Klein bottle. I mean having counterexamples helps to understand why they cannot be embedded.

In low dimension one may try to find such an example by saying we cannot immerse 3-ball into plane but that's trivial by definition (tangent space of any interior point of $B^3$ is 3-dimensional and $\mathbb{R}^2$ is 2-dimensional.)

In higher dimensions we have more interesting results on immersions of manifolds into Euclidean spaces. One should not think that every $n$-dimensional compact manifold can be immersed into $\mathbb{R}^m$ where $n<m$ because tangent space is lower dimensional, this is a wrong intuition or thinking.

For example, there is a compact $9$-dimensional projective space(manifold)($\mathbb{R}P^9$) and one cannot immerse it into $\mathbb{R}^{10},\mathbb{R}^{11}$ or even into $\mathbb{R}^{12}$. Lowest dimensional Euclidean space $\mathbb{R}P^9$ can be immersed is $\mathbb{R}^{15}$. This is calculated by using characteristic classes and it is difficult to give intuition for them without talking about cohomology or vector bundles.

Lastly, for punctured tori, once you puncture it, you can deform it continuously so that it becomes intersecting thickened 2 circles(cylinders). Since during the deformation process surface does not cross itself, the deformation is an immersion of punctured tori. You can also do the same deformation for punctured higher genus surfaces but eventually you will find more circles intersecting each other. May be this helps to see when a surface can be immersed into plane. Note that you cannot do such deformation to a closed surface.