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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: enter image description here

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?

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    $\begingroup$ The phrase "same global structure" is a reference to their topology. Your response that they "look very different in 3D space" is a reference to their geometry. They have the same topology, but very different geometries. $\endgroup$
    – RBarryYoung
    Mar 12, 2021 at 13:56
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    $\begingroup$ I voted to migrate this to Math.SE. $\endgroup$
    – AccidentalFourierTransform
    Mar 12, 2021 at 14:01
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    $\begingroup$ I'm no topologist, but the way I think of it is... If both shapes are made of thin rubber and you inflate them, they could both be spherical. This isn't true of all shapes, for example a donut. $\endgroup$
    – James
    Mar 12, 2021 at 18:00
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    $\begingroup$ It's funny that you accompany your question by an image that pretty much answers it: You have a 1:1 mapping between points on the two surfaces which preserves neighborhood relations. $\endgroup$
    – Peter - Reinstate Monica
    Mar 12, 2021 at 18:21
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    $\begingroup$ This is not a question about physics. $\endgroup$
    – JimmyJames
    Mar 12, 2021 at 19:07

2 Answers 2

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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.

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    $\begingroup$ This is not true. Topology studies continuous maps, which are notoriously different from smooth maps. On arbitrary topological spaces it doesn't even make sense to talk about differentiable maps. Also two spaces "having the same number of holes" is not equivalent to them being diffeomorphic (not even homeomorphic, i.e. equivalent in the sense of topology). I find it a bit dubious to post an answer to something of which you don't have a full grasp of. $\endgroup$
    – Jannik Pitt
    Mar 12, 2021 at 14:30
  • $\begingroup$ I don't see the reason to address an issue in this manner. My topology is a bit rusty but essentially there is issue, and that is with isometric with a plane being different from a sphere. You can't map a plane to sphere because of intrinsic curvature, you will get wrinkles or pinch points. Think the idea is related to betty number and cause curvature, and the central idea that a shape is classified by the number of holes or loops. $\endgroup$
    – marshal craft
    Mar 12, 2021 at 14:47
  • $\begingroup$ Also I think in math, and differential topology, global structure is a complicated subject, usually for people fairly well rooted in differential manifolds. It's probably best to consider it as rambling from the author, which is not uncommon. Basically its unclear, there isn't anything really to be learnt and similar is this "smooth crayon". $\endgroup$
    – marshal craft
    Mar 12, 2021 at 14:50
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    $\begingroup$ @JannikPitt To be fair, for closed surfaces there's essentially no distinction between the smooth category and the topological category (this is of course dramatically false in higher dimensions), and the classifying invariant is (give or take orientability) exactly the "number of holes" (i.e. the genus or, if you prefer, the rank of $H^1$) $\endgroup$
    – Denis Nardin
    Mar 12, 2021 at 15:23
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    $\begingroup$ @JannikPitt where is your answer? $\endgroup$
    – Greg Schmit
    Mar 12, 2021 at 15:32
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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.

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