gt.geometric topology - What is an immersed submanifold?

Bing's house with two rooms is the image of an immersed sphere that is not in general position.

General position immersions are easy to build out of local pictures --- well sort of easy to build. Consider inside an $n$-ball

$D^n=$ ${ (x_1,ldots, x_n) : sum x_j^2 le 1 }$ all of the $k$-dimensional sub-spaces that have
$(n-k)$ of the $x_j =0$. This is the local picture for a minimal dimension multiple point. (The greater the multiplicity, the smaller the dimension of the intersection). You don't have to choose all such subspaces, but only some of them. In this way you have local pictures to patch together. Now if you know how to attach handles to spaces, you can attach handles that have immersed pieces together. One can construct Boy's surface from this point of view.

Sometimes you get stuck. For example, a figure 8 has one double point. Boy's surface has one triple point. Capping of a generic sphere eversion gives a $3$-manifold in $4$-space with one quadruple point. But if you start from the intersection
$(a,b,c,d,0)$ $cap (a,b,c,0,e)$ $cap (a,b,0,d,e)$ $cap (a,0,c,d,e)$ $cap(0,b,c,d,e)$ in the $5$-ball, there is no way to close this off to get a $4$-manifold with one quintuple point. There are plenty of codimension $1$ immersions in $5$-space, but they all have an even number of quintuple points.

You should also consider equatorial spheres in a large dimensional sphere. This is the boundary of the second example I gave. You can connect these with handles to get connected immersions.

A very cool example in 3-space (beyond Boy's surface and an acme Klein bottle) is obtained by twisting a figure 8 a full rotation. A half a twist gives a Klein bottle, a full-twist gives an immersed torus whose stable framing is induced by the Lie group structure.

Codimension $0$ examples are also very important. The standard $2$-disk with two handles that represents a punctured torus is the image of an immersion into the plane.