My question begins with a caveat: I sometimes spend time with topologists, but do not consider myself to be one. In particular, my apologies for any errors in what I say below — corrections are encouraged.

My impression is that manifold topologists like to consider three main categories of (finite-dimensional, paracompact, Hausdorff) manifolds, which I will call $\mathcal C^0$, $\mathrm{PL}$, and $\mathcal C^\infty$, corresponding to manifolds whose atlases have transition functions that are, respectively, homeomorphisms, piecewise-affine transformations, and diffeomorphisms. The latter two categories obvious map faithfully into the first, and a theorem of Whitehead says that every $\mathcal C^\infty$ manifold admits a unique PL structure.

These categories are not equivalent in any reasonable sense. The generalized Poincare conjecture is true in $\mathcal C^0$, true (except possibly in dimension $4$) in $\mathrm{PL}$, and false in many dimensions including $7$ in $\mathcal C^\infty$. $\mathcal C^0$ is the realm of surgery and h-cobordism. In $\mathcal C^\infty$, and in particular in $4$ dimensions, there is a powerful tool called "gauge theory", which provides the main technology used to prove examples of homeomorphic but not diffeomorphic manifolds.

By definition, *gauge theory* is that part of PDE that studies connections on principal $G$-bundles for Lie groups $G$. The most important gauge theories for distinguishing between the $\mathcal C^\infty$ and $\mathcal C^0$ worlds are Donaldson Theory (which studies the moduli space of $\mathrm{SU}(2)$ connections with self-dual curvature) and the conjecturally equivalent Seiberg–Witten Theory (which studies an abelian gauge field along with a matter field, and which I understand less well). Another important gauge theory that I understand much better is (three-dimensional) Chern–Simons Theory, whose PDE picks out the moduli space of flat $G$ connections; for example, counting with sign the flat $\mathrm{SU}(2)$ connections on a 3-manifold is supposed to correspond to the Casson invariant.

My impression, furthermore, has been that the categories $\mathcal C^\infty$ and $\mathrm{PL}$ are in fact quite close. There are more objects in the latter, certainly, but in fact many of the results separating $\mathcal C^0$ from $\mathcal C^\infty$ in fact separate $\mathcal C^0$ from $\mathrm{PL}$. A side version of my question is to understand in better detail the distance between $\mathcal C^\infty$ and $\mathrm{PL}$. But my main question is whether the technology of *gauge theory* (possibly broadly defined) can be used to separate them. A priori, the whole theory of PDE is based on smooth structures, so it would not be unreasonable, but I am not aware of examples.

Are there gauge-theoretic invariants of smooth manifolds that distinguish nondiffeomorphic but $\mathrm{PL}$-isomorphic manifolds?

Of course, a simple answer would be something like "For any $X,Y \in \mathcal C^\infty$, the inclusion $\mathcal C^\infty \hookrightarrow \mathrm{PL}$ induces a homotopy equivalence of mapping spaces $\mathcal C^\infty(X,Y) \to \mathrm{PL}(X,Y)$." This would explain the impression I have that $\mathcal C^\infty$ and $\mathrm{PL}$ are close — if it is true, it is not something I recall having been told.