Gauge theory for embedded surfaces, II

10 P. B. Kronheimer and T. S. Mrowka
(iii) Independence.
The compactness result justifies the definition (2.4) for the invariants. The following
result summarizes some of the properties of this definition.
Theorem 2.13. As long as α remains in an interval of compactness for ν, the
invariant qk,l(u1,...,ue) defined by (2.4) is independent of α, ν and the choice of
metric in Cν . It is independent of the choice of the si and depends on the surfaces
ui only through their homology classes in X. As a function of these classes, qk,l
is multi-linear and symmetric; its degree e is given by the formula (2.1). Thus
qk,l is an invariant of acceptable pairs (X,Σ) and is a symmetric multi-linear
function
qk,l : [Σ] ⊥ ] ⊗e → Z,
where [Σ] ⊥ is the image of H2(X\Σ) in H2(X) (the complement of [Σ] determined
by the intersection form).
Proof. Since the proofs are little different from the arguments given elsewhere
(see [D4] or [DK]) we shall be brief, concentrating on the points which do not
have an immediate counterpart in the standard theory.
The proof that qk,l is independent of the choice of metric g ν within the space
C ν is quite familiar, following [D4] for example, and using the transversality
results for the twisted connections described in section 5(iii) of [KrM].
That qk,l is independent of the choice of the si and depends linearly on the
homology classes of the ui in X\Σ is also something which can be proved with
no essential modification of the usual argument; indeed, the proof presented in
[DK] can, like the compactness arguments, be simplified here, since the trivial
connection is absent from all our moduli spaces (see [DK], Theorem (9.2.12),
and the paragraphs following). The theorem, however, contains the assertion
that only the image of [ui] inH2(X) is material. The proof of this is fairly
elementary but requires some discussion of the topology of the configuration
space B∗ k,l ; we therefore postpone it to Appendix 1; see Proposition A1.6.
Now let Iε be an interval of compactness for ν and let α1 and α2 be in Iε. To
show that the values of qk,l computed using these two values of α are equal, recall
from [KrM], section 2(ii), that there is an extended moduli space ˆ Mk,l which, for
a generic choice of metric in Cν , is a smooth manifold of dimension 2e +1; there
) whose fibres are the moduli spaces
is a smooth mapping π from ˆ Mk,l to (0, 1
2
M α k,l . We can choose the metric so that all the moduli spaces ˆ Mk ′ ,l ′ are smooth
and free of reducible connections, and we can ask the same of the moduli spaces
ˆM α1
k,l and ˆ M α2
k,l . This means that α1 and α2 are regular values of π and that
π −1 [ α1,α2] is a smooth (2e + 1)-manifold with boundary.
Choose the sections si so that they vanish transversely on the extended
moduli space; the cut-down moduli space
(π −1 [α1,α2]) ∩ V1 ∩ ...∩Ve
is then a smooth 1-manifold with boundary. If we also arrange that all multiple
intersections of the Vi with the moduli spaces ˆ Mk ′ ,l ′ are transverse, then the same