I find this question interesting, but need to get it out of my system: is the space of connections (modulo gauge) on a compact four-manifold paracompact, in the Sobolev topology?


If so, I believe it would admit partitions of unity, which would surely make life easier in gauge theory. But I haven't seen the experts make use of such a fact. I have also heard that spaces of curves (as used in symplectic geometry) do not always admit partitions of unity.


The question occurred to me while reading about the first of the "five gaps" described by the late Abbas Bahri:
<http://sites.math.rutgers.edu/~abahri/papers/five%20gaps.pdf>


As Bahri himself points out, this so-called "gap" has been filled in several different ways (via the Freed-Uhlenbeck theorem or via holonomy perturbations; that is, if the original approach is indeed flawed). Still, I think it would be useful for the younger generation to understand the core of the difficulty. I had wondered if it can be summed up in a negative answer to this question.



Is the space of connections modulo gauge equivalence paracompact?
Yes, the space of gauge orbits of connections is paracompact (even when you the use Fréchet topology).


First, the space of all connections is paracompact since it is an affine space modelled on a nuclear Fréchet space (and/or it is metrisable). Narasimhan & Ramadas (Geometry of SU(2) Gauge Fields) showed that the action of the group of gauge transformation is proper (they actually only consider SU(2) gauge theories over $S^3$ but their argument generalizes to arbitrary structure groups and arbitrary compact base manifolds). For proper group actions, the orbit space is Hausdorff and the canonical projection map is closed. Since the [image of a paracompact Hausdorff space under a closed continuous map is also paracompact](https://ncatlab.org/nlab/show/Michael%27s+theorem), we conclude that the orbit space is paracompact.


Moreover, the gauge orbit space is stratified by smooth (Fréchet) manifolds (it's the usual orbit type stratification, the top stratum being the space of irreducible connections, see for example [The orbit space of the action of gauge transformation group on connections](http://dx.doi.org/10.1016/0393-0440(89)90025-9) or [On the gauge orbit space stratification: a review](http://iopscience.iop.org/article/10.1088/0305-4470/35/28/201/meta)). Since every stratum is modelled on a nulear Fréchet space, they are even *smoothly* paracompact (see Convenient Setting of Global Analysis by Kriegl & Michor for further details, especially chapters 14 (Smooth Bump Functions) to 16 (Smooth Partitions of Unity and Smooth Normality)).


