Why Nonmanifolds are Bad
An n-dimensional manifold (n-manifold for short),
is a space which locally, in the vicinity of each point, looks like
n-dimensional Euclidean space. In particular 2-manifolds are known
as surfaces, which locally look like the Euclidean plane.
A well-known example is a sphere, such as the surface of the earth.
(For this reason it was long believed that the earth was flat.)
Why are manifolds particularly noteworthy? Well let's look at something
which isn't a manifold:
where the abstract gluings are indicated by the labellings and arrows on the
edges. At first glance this looks like a perfectly comfortable place for our
2-dimensional crab to live in, no worse than a Klein bottle.
However appearances are deceiving. Let's see what happens when the crab
approaches an edge which is abstractly glued to two other edges:
Here is another local view of the situation, equivalent from the crab's
point of view:
Where will the crab reemerge when it passes through the edge where the
three sheets meet? It will have to reemerge in one of the other two sheets,
but which one? There is no good way to answer this question, so it seems
that the crab will emerge in either of the other two sheets with equal probability.
(The crab can't control where it will go, eg. what happens if the crab just
coasts ahead on its present course?) This is quite disturbing already, but
the actual situation is much worse for the crab. For the crab is not an indivisable
point particle. Rather it is composed of (2-dimensional) atoms. Each of these
component atoms has an equal probability of reemerging in either of the two
sheets. Thus on average half of the crab's atoms wind up in one of the two sheets,
the the other half in the other sheet. Thus the end result will be (very) finely ground