Study topology from existence of multiple smooth structures?

Question

There are classical existence results of a smooth structure on a topological manifold, and many results on the existence of multiple (i.e. exotic) smooth structures. Some utilize Freedman's theorem in dimension 4, and there's Milnor's twisted spheres in dimension 7, etc..

Morse theory and Riemannian geometry, for example, help to probe topology and only need a smooth structure. I questioned why we would want to know whether there is more than one smooth structure, and it leads to a less subjective question: Are there ways to study the underlying topological manifold from the existence of multiple smooth structures?

Here is a more refined question: Can different smooth structures give you different "bounds" on some topological quantities (maybe minimal genus of surfaces, "optimal" presentations of the fundamental group, etc.), or where one smooth structure gives you information while the other smooth structure can't give you any?

Answer 1

Since this question might have many answers, I can propose one possible answer (I hope this is really an answer to the question.) So, suppose we have a smooth $4$-dimensional manifold. We want to triangulate it smoothly so that the number of vertices is the minimal possible. Then this minimal number of vertices might well depend on the choice of a smooth structure. Indeed, if we take a smooth compact $4$-fold that has infinite number of smooth structures, then for any $N$ there will be only finite number of such structures for which there exists a triangulation with at most $N$ vertices.

It is well known that for $\mathbb C \mathbb P^2$ with the standard smooth structure the minimal number is $9$ . Such a triangulation was invented by Kühnel (and some other people, see references in the article below). If by any chance $\mathbb C \mathbb P^2$ has an infinite number of smooth structures, there will be those for which the minimal number of vertices is huge (the number is always $\ge 9$).

By the way, there is also a triangulation of a K3 surface (also constructed by Kühnel with a collaborator) with 16 vertices. This result uses Freedman's theorem. As far as I understand, it is not yet known whether the corresponding K3 surface is diffeomorphic to the standard one, see

https://www.sciencedirect.com/science/article/pii/S0040938399000828