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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.

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2 Answers 2

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First, it follows from the work of Kirby and Siebenmann that in dimensions $\le 6$ PL and DIFF categories are equivalent. In particular, if you are working in dimension 4 (where gauge-theoretic invariants are mostly used) then the answer to your your question is negative. Starting in dimension 7, there are smooth manifolds which are PL-equivalent but not diffeomorphic. Milnor's exotic 7-spheres are the first examples of such manifolds. The smooth structures on Milnor's spheres are distinguished via index and 1st Pontryagin class. Whether you consider such invariants gauge-theoretic or not, depends on how broadly you interpret gauge theory. For instance, characteristic classes of smooth manifolds can be defined via differential forms, i.e., Chern-Weil theory, or as indices of some elliptic operators. Does this qualify as gauge theory? (You can lift forms from, say, tangent bundle to the principal bundle- frame bundle, if you so desire.) The point of usage of gauge theory in dimension 4 is that the "traditional" topological invariants turned out to be insufficient, so one considers spaces of connections satisfying some differential equation (like self-duality) and uses such spaces to derive some smooth invariants of 4-manifolds. As far as I know, nobody used this viewpoint in higher (i.e., at least 7) dimensions, since there was no need for it.

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In Foundational Essays on Topological Manifolds, Smoothings, and Triangulations, Kirby and Seibenmann show homotopy equivalence of the Kan complexes $\mathrm{Man}^m_{sm}$ and $\mathrm{Man}^m_{PL}$ for $m\leq 3$, , that can be thought of as classifying spaces of smooth and of PL $m$-manifolds correspondingly (an earlier version of this answer stated this for all $m$, which is false). Jacob Lurie wrote a very nice set of lecture notes on just this topic. Perhaps this is the statement you want? Restricted to one skeletons, it gives homotopy equivalence of mapping spaces for low dimensions- but the full statement tells you much more, and it provides a precise and intuitively satisfying sense in which smooth and PL categories are `close' for low dimensions. In higher dimensions, PL manifolds can be smoothed in dimension $\leq 7$, essentially uniquely in dimension $\leq 6$.

The definitions of the `classifying spaces' are roughly as follows. For a finite dimensional vector space $V$ and for a smooth $m$-manifold $M$, the simplicial set $\mathrm{Emb}^m_{sm}(M,V)$ is defined to have $n$-simplices as embeddings $M\times\Delta^n\rightarrow V\times\Delta^n$ which commute with the projection to $n$. Now let $\mathrm{Sub}^m_{sm}(V)$ denote the simplicial set of submanifolds of $V$, whose $n$-simplices are given as smooth submanifolds $X\subseteq V\times\Delta^n$ such that the projection $X\rightarrow \Delta^n$ is a smooth fibre bundle of relative dimension $n$. If $V$ is infinite dimensional, we define $\mathrm{Sub}_{sm}(V)$ as the direct limit of $\mathrm{Sub}_{sm}(V_0)$ as $V_0$ ranges over all finite dimensional subspaces of $V$. The parallel definitions in the PL case define $\mathrm{Sub}_{PL}(V)$. For a fixed infinite dimensional vector space $V$, the Kan complexes $\mathrm{Man}^m_{sm}$ and $\mathrm{Man}^m_{PL}$ are defined as $\mathrm{Sub}^m_{sm}(V)$ and $\mathrm{Sub}^m_{PL}(V)$ correspondingly.

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(I fixed a formatting conflict.) –  Theo Johnson-Freyd Mar 17 '13 at 17:37
2  
Kirby and Siebenmann show no such thing, as the statement claimed is false. The $E_8$ PL-manifold in any dimension $4k \geq 8$ (obtained by forming the $E_8$-plumbing and coning off the boundary, which is a PL sphere) is not smoothable, so any reasonable map $Man_{sm}^m \to Man_{PL}^m$ is not even surjective on path-components, never mind a homotopy equivalence. –  Oscar Randal-Williams Mar 17 '13 at 21:31
    
@Oscar: corrected to make the statement only for $m\leq 3$. Sorry about that. –  Daniel Moskovich Mar 18 '13 at 1:22

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