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.