Is the surface of a sphere and a crayon the same manifold? In Schutz, Geometrical Methods of Mathematical Physics book, pg. 29, it was said that the sphere $S^2$ and the surface of a crayon has the same global structure:

I also read that two spaces are the 'same' as manifolds if they are diffeomorphic.
But I find it hard to visualise that these two surfaces have the same global structure. They clearly look very different in 3D space. What exactly does global structure mean?
 A: There is an old joke that a topologist is a mathematician who doesn't know the difference between a donut and a coffee cup.
Topology considers them the same because they are both closed surfaces that have one hole in them. Likewise, a crayon and a sphere have no holes.
The actual shape doesn't matter. topology studies the properties of an object that stay the same when the object is smoothly distorted. Tearing or poking holes is not allowed. Number of holes is such a property.
Such a smooth distortion allows you to map points from the original shape onto the final shape. Every point will be near the same points it was near in the original shape. You can reverse the distortion. Each point will return to its original location. The mapping is reversible.
A: Geometry is taken to be the study of distances and angles, hence the study of the metric. If we forget the metric, we still have what is called the smooth structure, and if we forget the smooth structure, we still have the topological structure.
Whilst the surface of a crayon is not metrically equivalent to the surface of a sphere, they are smoothly equivalent, and hence also, topologically equivalent.
Smooth equivalence, roughly speaking, allows deforming the object through squeezing and stretching so long as no creases or folds are introduced and furthermore no cuts.
Imagine the crayon was made out of plasticine, then we can easily see that we can deform this crayon to a ball, simply by squeezing it and rolling it up into a ball, and hence the surfaces are diffeomorphic. Its also notable here, that the bodies of the two objects and not just their surfaces are diffeomorphic. This need not be the case, for example a hollow sphere and a ball have the same surface and so are diffeomorphic, but the hollow sphere has no interior whereas the ball does, so their bodies are not diffeomorphic.
