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In General Topology if we a topological space $(X, \mathcal{T})$, that is homeomorphic to another space $(Y, \mathcal{K})$, there are a number of topological properties, such as compactness, connectedness, path-connectedness, that are invariant under homeomorphism.

In Algebraic Topology if we have homeomorphic topological spaces $(X, \mathcal{T})$ and $(Y, \mathcal{K})$, then we can conclude that they have isomorphic fundamental groups $\pi_1(X) \cong \pi_1(Y)$

But in Differential Topology, the question of what topological properties are preserved by diffeomorphisms seems to be something that I can't quite answer at the moment.

Certainly diffeomorphisms are stronger versions of homeomorphisms, so all the things we expect to be invariant under homeomorphisms (compactness, connectedness etc.) are also invariant under diffeomorphism.

However I would like to know if there are further topological (or perhaps non-topological) properties that are invariant under diffeomorphism?

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Your question has two parts:

Are there further topological properties that are invariant under diffeomorphism?

The straightforward answer is this: Diffeomorphisms preserve exactly the same topological properties as homeomorphisms do; nothing more, nothing less.

The reason for this is essentially definitional: A topological property is, by definition, a property that is preserved by homeomorphisms. Since every diffeomorphism is a homeomorphism, every topological property is preserved by diffeomorphisms. And if a particular property of smooth manifolds is preserved by diffeomorphisms but not by homeomorphisms, then it's not a topological property.

The other half of your question was

Are there further non-topological properties that are invariant under diffeomorphism?

This is a more interesting question. But first you have to establish an appropriate category of spaces to work in -- diffeomorphisms are only defined between smooth manifolds, which are topological manifolds with an additional structure called a smooth structure. So the appropriate question to ask is whether there are non-topological properties of smooth manifolds that are preserved by diffeomorphisms. There are, but they're more subtle. For example, one such property for compact smooth manifolds is whether they bound (smoothly) parallelizable manifolds. This is one way that exotic spheres can be distinguished from each other.

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  • $\begingroup$ Wow, I can't believe its you, thank you for this concise answer. I'm currently reading your Introduction to Smooth Manifolds book, and it is a complete joy to read and work through! $\endgroup$ – Perturbative Jul 28 '17 at 19:19
  • $\begingroup$ @Perturbative: Glad to hear it! Thanks for the kind compliment. $\endgroup$ – Jack Lee Jul 30 '17 at 0:47
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It isn't true that spaces with isomorphic fundamental groups are homeomorphic. For example, the circle and the infinite cylinder. These both have fundamental group $\mathbb{Z}$, yet one is compact and one is not.

Anyway, putting that aside, the real answer to your question is that homeomorphisms can be arbitrarily squiggly, while diffeomorphisms, needing to be differentiable, can only be so squiggly. What this means is that local properties are going to be preserved by diffeomorphism that are not preserved by homeomorphism. One example of this is the Hausdorff dimension of a space, like a Koch snowflake or a Julia set. If you take some self-diffeomorphism of the plane, the Hausdorff dimension of the image of the fractal will be the samea s the Hausdorff dimension of the original fractal. Homeomorphisms certainly won't do this though! Something like the Koch snowflake will be homeomorphic to a circle!

A good way to find other examples is by remembering that diffeomorphisms only make sense in spaces that allow us to talk about differentiability, i.e. are in some sense Euclidean. So if we just think about ordinary metric spaces, we can find an easy example - Cauchy sequences. Most metric spaces aren't complete. Homeomorphisms will preserve the property of convergence, but since the space may not be complete, we can have missing limit points. The easiest example I can think of is something like the sequence $1/n$ in the point being mapped to something like $(0, 1/n)$ in hyperbolic space, where the distances between points goes to infinity as $y \to 0$. We can make hyperbolic space locally Euclidean using the concept of differentiable manifold though, but I think the point is clear thinking of hyperbolic space in the sense of synthetic geometry.

However you can step up from diffeomorphism to things like isometry when dealing with Riemannian manifolds, which also have a great many properties that they preserve, which are not visible in topological manifolds at all. For example, curvature, geodesics, and a host of other intrinsic properties.

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    $\begingroup$ To elaborate on the first point for the OP: $\pi_1$ is invariant under homotopy equivalence, not just homeomorphism. It's also nowhere close to completely classifying spaces modulo homotopy equivalence; there are lots of spaces with $\pi_1 = 1$, for example (e.g., any universal cover). Even if you include the higher $\pi_i$, it only classifies spaces modulo homotopy equivalence in the CW-complex category and with the assumption that the isomorphisms $\pi_* X \to \pi_* Y$ are actually induced by a map, not just abstract isomorphisms. $\endgroup$ – anomaly Jul 27 '17 at 16:46
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"However I would like to know if there are further topological (or perhaps non-topological) properties that are invariant under diffeomorphism?"

Yes. If you consider diffeomorphisms, the differential structure itself is an invariant. Look here : https://en.wikipedia.org/wiki/Exotic_R4

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One of the harder theorems about manifolds is Novikov's 1966 theorem that the Pontryagin classes of a smooth manifold, which had already been well understood as diffeomorphism invariants for a couple of decades before, were actually invariant under homeomorphisms as well.

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