Why smooth manifolds?

Question

An analytic structure on a manifold is an atlas such that all transition maps are real-analytic.

I am wondering firstly about times where a real manifold might arise where the full generality of a topological manifold is required, as opposed to a smooth or analytic one. Secondly I am wondering about times where it would be better to only consider the smooth structure.

Personally I find it something of a nuisance that not all topological manifolds can be triangulated. To me this is a sign of a definition which is unnecessarily general. But since I am no expert in this area, I am wondering what some uses might be or if people find reason to object.

Edit: I'd like to specify that I'm interested in examples of intrinsic interest where a topological manifold arose- not so much a distaste for topological manifolds.

Edit: It's normal to have opinions about proper mathematics, but as for this question about situations when purely topological manifolds arise, we would do well to ensure that these opinions not prevent the question from expression or distract from best-effort inquiry.

Answer 1

For $n \not= 4$, a smooth manifold homeomorphic to $\mathbf R^n$ is diffeomorphic to $\mathbf R^n$, so $\mathbf R^n$ as a topological manifold has just one smooth structure. The story is totally different for $n = 4$: look up exotic $\mathbf R^4$. There are $28$ different smooth structures on a $7$-dimensional sphere. That is, if you consider smooth manifolds that are homeomorphic to the smooth manifold $S^7$ but not necessarily diffeomorphic to it, there are $28$ examples. Look up exotic spheres.

In describing such phenomena, you could choose to talk about manifolds that are homeomorphic but not diffeomorphic or you could choose to talk about different smooth structures on a specific topological manifold. The choice is up to you.

Answer 2

Here is an answer from the dynamics point of view, which seems relevant to the discussion as arguably one of the reasons for defining manifolds was to formalize the notion of a "phase space at large" (see e.g. Mackey's The Mathematical Foundations for Quantum Mechanics, p.10 for a discussion of this). Note that this is in a sense a reflection of the categorical motto that "arrows are more important than objects" (e.g. at https://plato.stanford.edu/entries/category-theory/ this is attributed to Eilenberg & MacLane).

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Let us consider the problem of classification of $\mathbb{Z}$-actions on (smooth) manifolds via diffeomorphisms; thus we have a manifold $M$, a diffeomorphism $f:M\to M$ of it, and the dynamical system at hand is the iterates $...,f^{-1},\text{id}_M,f,f^2=f\circ f,...$. In a sense we would like to say that $f:M\to M$ and $g:N\to N$ are equivalent if the behavior of $x\in M$ under $f$ is the same as the behavior of some $y=y(x)\in N$ under $g$. There are multiple ways of formalizing this, but a naive categorical approach points to an isomorphism in the category of smooth manifolds (and smooth maps) carrying a $\mathbb{Z}$-action, that is to say, $f:M\to M$ and $g:N\to N$ are the same if there is a diffeomorphism $\Phi:M\to N$ such that $\Phi\circ f=g\circ \Phi$.

It turns out that this is too much too ask from the dynamical point of view. E.g. the systems $x\mapsto 2x$ and $x\mapsto 3x$ on $\mathbb{R}$ are not isomorphic in this sense, but for either system there is exactly one fixed point and all other points escape to infinity exponentially fast as time goes to $+\infty$, so that in a certain sense they are the same (trying to compare the systems $x\mapsto cx$ and $x\mapsto dx$ with $c$ and $d$ known to be equal only to a certain number of digits highlights a situation where the latter perspective could be more natural); more generally, assuming that there exists periodic points (and under mild conditions there are periodic points), the eigenvalues of the derivatives of the systems become smooth conjugacy invariants; they actually become complete smooth conjugacy invariants in certain circumstances (after the work of de la Llave-Marco-Moriyón and others).

Thus one asks for a worse coordinate change, and after a diffeomorphism (and a bi-Lipschitz homeomorphism) the next best kind ($\star$) of a coordinate change is a homeomorphism (note that this works for the above example; though of course one does not have uniqueness of the conjugating homeomorphism). And indeed this perspective has been quite fruitful (see also Smale's survey "Differentiable Dynamical Systems", pp.748-749). So it becomes important to keep track of what structure is being respected by the isomorphism of the dynamical systems. Surely a homeomorphism between manifolds preserve the topological structure (i.e. the notion of proximity), but even more, they preserve the local Euclidean structure (sans differentiability). In this sense it becomes important to take the change of coordinates not as an isomorphism in the category of topological spaces (and continuous functions), but in its full subcategory of topological manifolds.

An instance of this is a classical theorem by Anosov, which says that if a diffeomorphism $f:M\to M$ on a closed smooth manifold is "uniformly hyperbolic", then for any diffeomorphism $g:M\to M$ sufficiently close to $f$, there is a homeomorphism $\Phi:M\to M$ such that $\Phi\circ f=g\circ \Phi$ (see Examples of conjugate-like structures across mathematics for the definition of a uniformly hyperbolic diffeomorphism). The homeomorphism involved will be always Hölder, but except on rare occasions it will not be smooth (see also Definition of Hölder Space on Manifold). Anosov also constructed an example of a real-analytic Anosov diffeomorphism (on a real-analytic manifold) such that the conjugacy fails to be even continuously differentiable.

On the other hand, it is not always the case that the isomorphism is always a homeomorphism; indeed sometimes the isomorphism is as regular as the dynamics (which is called "smooth rigidity", broadly speaking), thus it becomes important to have separate accounts for different hierarchies of structures on manifolds.

There are also cases where the dynamics is, at least at the syntactic level, known but what (local/global) structure it preserves is unclear; in such cases too having separate accounts becomes valuable.

($\star$) One can and does ask for even worse coordinate changes, e.g. a measurable map with measurable inverse (hopefully with some more properties at least in certain special directions defined by the dynamics).

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In closing here are similar discussions to which I've made humble contributions with further examples and references, all from a similar point of view: Motivations of studying $C^r$ manifolds when $r<\infty$, What is the relationship between the vector fields of conjugate flows?, Why Do We Care About Hölder Continuity?. Regarding the history my answer at Why is homeomorphism understood as stretching and bending? seems relevant.