I am attempting to extrapolate a 3D case to 4D. I hope it's alright to use some physics terminology. 2+1 space = 2 spatial dimensions and 1 time dimension. World-line/sheet/volumes are simply the objects traced out as the spatial dimension progresses through time.
Suppose you have a 2+1 space with a stationary particle tracing out a world line. Now imagine that for a period of time you take a line segment constrained to the 2-space and form it into a circle around that particle. You then break the circle and remove the line segment. When one looks at the world volumes this will result in a 'threading' action where the the particle world-line is threaded through a hole in the line segment world-sheet.
Now my question is, when we upgrade to 4D. i.e. when we temporarily surround a stationary particle in 3D by forming a 2D surface into a sphere, and then break that sphere and remove the 2D surface, do we obtain the same kind of 'threading' action as we saw in the previous case.
Any advice as to what formalism could be used to examine such things would be much appreciated.
Thanks!
In the picture you drew, the point traces out a vertical line and the line/circle in each slice traces out a surface. I can see two ways to read your picture which give slightly different surfaces.
The first way to read your picture is that the vertical line on the right of the point in the first picture remains there no matter how high we go and that the vertical line on the right of the point in the last picture is there no matter how low we go. In this case, the surface traced out is a non-compact surface with two boundary components.
The second way to read your picture is that the vertical lines in the first and last pictures disappear to a point and then to nothing at some height (both up and down). In this case, the surface we've traced out is just a cylinder.
In one higher dimension, basically the same thing happens. Each slice is now a $3$-dimensional picture. In the initial slice, we have a point together with a surface with boundary. However, here we have quite a bit more choice than in $2+1$-dimensional case, and that choice is a choice of surface. Our surface could be a $2$-dimensional sphere or a connected sum of $g$ different tori (called the genus $g$ surface). See here for more details. For simplicities sake, I'm not considering the non-orientable surfaces.
The "easiest" surface to consider would be the sphere, and here the story unfolds like the $2+1$-dimensional case. The sphere has a small "hole" in it where the boundary is. The boundary swallows the point and closes into a sphere. Later a new boundary curve opens up and lets out the point.
In the other cases, the same sort of thing happens, except there is one new wrinkle. Let's take the torus for example. The torus starts with a boundary component which swallows the point. Then the boundary component closes to give a closed torus with a point inside. However now the path of the point matters. The point can travel around the torus in loops in either direction some number of times before the boundary component opens up to let the point out again. Different choices of paths would be topologically distinct configurations. For a higher genus surface, the analysis looks much like the torus case, except there is even more choice of path for the point to follow.
I'm not sure if this is the type of description you are looking for, and I'm also not sure how any of this can be used. I hope my description has been helpful in some way.