Gauge theory for embedded surfaces, II

Gauge theory for embedded surfaces, II 45
Lemma 7.6 with the metric g(s). Suppose that α(s) is less than both α1(s) and
α0 for all s, and that α(s) remains in the interval I. Then the value of Γ is
constant in the family.
Proof. As usual, we can form the parametrized moduli space and its compacti-
fication. The map ¯ρ extends continuously to the parametrized moduli space, so
even the fundamental class ¯ρ∗[ ¯ M α 0,l ] is unchanged in the family. ⊓⊔
Remark. The point here is that, a priori, two metrics satisfying the conditions
laid down need not be connected by any path of the sort asked for in the lemma.
To tidy up the situation, it would be necessary to consider extending Γ to an
invariant in the sense of [D2]. That is, one should consider the (α, t) plane to
be divided into chambers by the walls given in 7.4, and one should hope to
understand how Γ will change as a path (α(s),g(s)) crosses one of the walls.
On the basis of this, one should seek to show that Γ was a function only of the
chambers. As it is, we have considered only one chamber (the one with α small),
andwehavenotshownthatΓ takes only one value here.
(iii) Relating Γ to p0,l.
Consider forming the ruled surface X as the union of manifolds with cylindrical
ends, X o and W o . The latter space W o is, as usual, a complex line bundle of
degree n over Σ, while X o is a line bundle of degree −n over Σ ′ . The situation
is not symmetrical between these two however, since as usual we have nontrivial
holonomy and a cone-like metric along Σ ⊂ W o . Fix metrics with cylindrical
ends on both pieces, and join them together as usual with a neck of length
R. Note that the quantity α1 for the resulting metric gR can be taken to be
independent of R, since the geometry near Σ ′ is determined by the metric on
X o alone. Fix α less than both α1 and α0 as before, and take a suitably large ν.
Using these values of α and ν and the metric gR, obtain an integer Γ as
above. By Lemma 7.9 the result is independent of R. From the manifold W o
with its cylindrical end metric, obtain also the invariant p0,l. Then we have the
following counterpart to Theorem 5.10:
Proposition 7.10. For the metrics as above, the integer Γ is given by
Γ = p0,l × ∆,
where ∆ is the degree of the moduli space of flat connections inside Bo Σ ′; that is,
∆ = µ 3g−3 , [Mflat(Σ ′ )] .
Proof. Choose 3g − 3 sections of the standard line bundle on Bo Σ ′ so that there
common zero set V is transverse to Mflat and meets it in a finite number of points
in the irreducible part of the moduli space. The number of these points, counted
algebraically, will be ∆. We can also arrange that these points are regular values
for the map
rl : M α 0,l (W o ,Σ; R s + )→R+