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Some mathematical structures are visualized very well. I imagine how a shapeless bunch of points (a set; the only property of which is quantity) is collected in one or another soft form (topological space), depending on what topology is set on it. I imagine how this soft (floating) shape becomes hard when I set the metric. The first examples of deformation retractions give a vivid geometric intuition of the homotopy type as a "(homotopy) framework". Thus, the concepts of homotopy type, topology, and metrics have a clear physical content. In contrast to these three examples, for the concept of a smooth structure on a manifold, everything is not so clear due to the fact that manifolds up to dimension 3 have a unique smooth structure

Do you have any intuition for a smooth structure on a manifold? Do you see any meaning/physical content in it? Do visually homeomorphic non-diffeomorphic spaces differ for you?

After a smooth manifold is provided with additional structures (Riemannian, symplectic, etc.), the meaning of these objects becomes clear. But a smooth manifold devoid of additional structures is still very mysterious for me.

P.S. I'm not sure if this question is for a forum (on the other hand, the system showed me many questions with similar titles). If so, then feel free to close it.

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    $\begingroup$ Is it that different from "visualizing" a function (of one real variable, say) which is continuous but not differentiable? $\endgroup$
    – Sam Hopkins
    Commented Mar 22, 2022 at 22:23
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    $\begingroup$ Probably yes, because the smoothness of a function is a property, and "the smoothness of a manifold" is an additional structure. At least continuous non-smooth functions themselves are easy to visualize, for example $|x|$. $\endgroup$
    – Arshak Aivazian
    Commented Mar 22, 2022 at 22:26
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    $\begingroup$ I think the question is not about the distinction between manifolds that have or don't have smooth structures, or between maps of manifolds that are or aren't smooth, but given a manifold that does have at least one smooth structure, what do the different ones mean. A related question is "what's the simplest example of two distinct smooth structures on a given topological manifold ?" $\endgroup$
    – Maxime Ramzi
    Commented Mar 22, 2022 at 22:30
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    $\begingroup$ @MaximeRamzi: maybe the two questions are not so separate, since whenever you have two smooth manifolds that are homeomorphic but not diffeomorphic, any homeomorphism gives a continuous, non-differentiable function. $\endgroup$
    – Marco Golla
    Commented Mar 22, 2022 at 22:41
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    $\begingroup$ As for the "simplest example", there are at least two candidates: 1. $3\mathbb{CP}^2\#20\overline{\mathbb{CP}}\vphantom{C}^2$ and $K3\#\overline{\mathbb{CP}}\vphantom{C}^2$; 2. $S^7$ and $\{v^2+w^2+x^2+y^3+z^7 = 0\} \cap S^9 \subset \mathbb{C}^5$. $\endgroup$
    – Marco Golla
    Commented Mar 22, 2022 at 22:45

1 Answer 1

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$\newcommand{\R}{\mathbb{R}}$The purpose of a manifold structure is to be able to differentiate functions. And initially we know how to differentiate functions only if the domain is an open set in $\R$. Moreover, once you know how to differentiate functions, this opens the door to doing all the other stuff you do with manifolds.

Here's how I think of it: Initially, think of a manifold as just a set $M$ and nothing else. Think of a coordinate map as a bijection $\phi: O \rightarrow \R^n$, where $O\subset M$ and $\phi(O)$ is an open subset of $\R^n$. An atlas is a collection of coordinate maps such that the domains of the coordinate maps cover $M$. No assumptions on topology or differentiability yet.

A topological manifold is $M$ with an atlas $\mathcal{A}$ such that for any two coordinate maps $\phi_1: O_1 \rightarrow \R^n$ and $\phi_2: O_2 \rightarrow \R^n$ such that $O_1\cap O_2\ne \emptyset$, then the map $\phi_2\circ\phi_1^{-1}: \phi_1(O_1\cap O_2) \rightarrow \phi_2(O_1\cap O_2)$ is a homeomorphism. Notice that such an atlas immediately defines a topology on $M$ where the domains of the coordinate maps in $\mathcal{A}$ form a base of open sets. Assume that this topology is Hausdorff, and you have a topological manifold.

A smooth manifold is defined in exactly the same way, except you assume that the change of coordinate maps, $\phi_2\circ\phi_1^{-1}: \phi_1(O_1\cap O_2) \rightarrow \phi_2(O_1\cap O_2)$, are smooth.

Clearly, a smooth manifold is a topological manifold. If you start with a topological manifold, extend its atlas to a maximal atlas, then you can ask whether there is a subatlas (i.e., a subset of coordinate maps) that satisfies the definition of an atlas of a smooth manifold. If so, you say that the topological manifold is smoothable.

There is no reason why there couldn't be two different smooth subatlases of a topological manifold. And, if there are two such subatlases, there is no reason why they should be compatible with each other. In other words, if you have a coordinate map $\phi_1: O_1 \rightarrow \R^n$ in the first smooth atlas and a coordinate map $\phi_2: O_2 \rightarrow \R^n$ in the second smooth atlas, it does not necessarily follow that $\phi_2\circ\phi_1^{-1}: \phi_1(O_1\cap O_2) \rightarrow \phi_2(O_1\cap O_2)$ is smooth. It is homeomorphic, since both lie in the topological atlas.

Now, to make things even more complicated, it is possible that there is a global homeomorphism $\Phi: (M,\mathcal{A}_1) \rightarrow (M,\mathcal{A}_2)$ that is a smooth diffeomorphism of the two apparently different smooth manifolds. So, even though the two subatlases are incompatible, they actually define two smooth structures on $M$ that are diffeomorphic.

Finally, we say that a topological manifold has more than one smooth structure on it if there are two smooth subatlases such that no map $\Phi$ as described in the previous paragraph exists.

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  • $\begingroup$ Oops. I forgot return to functions. You can now say that a function $f: M \rightarrow \R$, where $M$ is a smooth manifold, is smooth if $f\circ\Phi^{-1}: \phi(O) \rightarrow \R$ is smooth for any coordinate map $\phi$. The assumption on the change of coordinate maps is the necessary and sufficient condition that you don't run into any logical inconsistencies in this definition. $\endgroup$
    – Deane Yang
    Commented Mar 22, 2022 at 23:26
  • $\begingroup$ I understand this (and love the language of sheaves), but it does not answer the question of what physical idea the notion of smooth structure captures. Imagine two homeomorphic non-isometric metric spaces. You see how they differ. Imagine two homeomorphic non-diffmomorphic manifolds. Do you see how they differ? $\endgroup$
    – Arshak Aivazian
    Commented Mar 22, 2022 at 23:37
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    $\begingroup$ Alas, I didn't really answer the question you asked. Unfortunately, I don't see any physical intuition for two homeomorphic but non-diffeomorphic manifolds. Notice that this is a global concept. In general, there will exist diffeomorphisms between dense open subsets of the two manifolds that can be extended uniquely as homeomorphisms that are not diffeomorphisms. $\endgroup$
    – Deane Yang
    Commented Mar 23, 2022 at 0:23
  • $\begingroup$ MathJax note: As unpleasant as it is, you have to write $\newcommand\R{\mathbb R}$The purpose …, not $\newcommand\R{\mathbb R}$ The purpose …, or your text will have a spurious space. I have edited accordingly. $\endgroup$
    – LSpice
    Commented Mar 23, 2022 at 19:10
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    $\begingroup$ @LSpice, thanks! $\endgroup$
    – Deane Yang
    Commented Mar 23, 2022 at 20:29

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