Gauge theory for embedded surfaces, II

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$Math 126 Solution Set 2$

$Math S?101. Worksheet 1. Axioms of Integer Arithmetic$

44 P. B. Kronheimer and T. S. Mrowka Proof. By Uhlenbeck’s regularity results, there exist a constants κ0(N ′ )andC, depending on the geometry of a neighbourhood N ′ of Σ ′ ,suchthatif[A]isany anti-self-dual connection on N ′ with action less than κ0(N ′ ), then sup N ′|FA|≤CFA L 2 (N ′ ). In particular, if the action of A is sufficiently small, then the C0 norm is also sufficiently small to ensure that the integral of |FA| on Σ ′ is less than 2π, which in turn ensures that A cannot be reducible of non-zero degree. Since the total action of the connections in M α 0,l ′ goes to zero with α, the lemma follows. ⊓⊔ Note that, in particular, there are no reducible solutions in the moduli space if α
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+
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Gauge theory for embedded surfaces, II

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