Gauge theory for embedded surfaces, II

12 P. B. Kronheimer and T. S. Mrowka
3. Manifolds with cylindrical ends
(i) Flat connections on the 3-manifold.
We now wish to gather together and summarize some of the facts that we shall
need concerning instanton moduli spaces on manifolds with cylindrical ends.
Nearly all of this material is drawn from [T], [MMR] and [MM].
The complement of Σ in the closed tubular neighbourhood N is diffeomorphic
to the half cylinder R + × Y ,whereY is the circle bundle over the surface. Let
g o be a metric on N\Σ which makes it isometric to such a cylinder, and let g o
be extended smoothly to X\Σ in some way. We shall write X o for the manifold
X\Σ when we have in mind that it is equipped with such a cylindrical-end
metric.
Let E be an SU(2) bundle over X o (necessarily trivial on this open 4manifold),
let A, orA(X o ), be the space of smooth connections in E with finite
action,
A = { A | FA ∈ L 2 }, (3.1)
and let G be the group of smooth gauge transformations. The cylindrical-end
moduli space is then the quotient
M(X o )={A∈A|F +
A =0} G.
The space A is topologized so that a sequence Ai is convergent to a connection
A provided only that there is C∞ convergence on compact subsets and that no
action is lost in the limit:
lim |F (Ai)| 2
= |F (A)| 2 .
The moduli space inherits a topology as a quotient of a subset of A.
For each t>0, let Yt be the slice Y ×{t} of the cylindrical end. The first
substantial fact about the moduli space is the existence of a limit of the gauge
equivalence classes,
r(A) = lim A|Yt
t→∞
for every A in M(X o ). This defines a continuous map
rXo : M →R(Y) (3.2)
to the moduli space of flat connections on Y . We refer to R(Y ) as the ‘representation
variety’. Note that our terminology differs from [MMR] for example,
where R(Y ) denoted the space of homomorphisms from π1(Y ) to SU(2); our
R(Y ) is the quotient of that space by the action of SU(2) by conjugation.
We need a few facts about R(Y ), which we take from [T]. Henceforth we will
assume that the genus of Σ is at least 2 and that the self-intersection number
Σ·Σ is non-zero. We shall write n for Σ·Σ.