Gauge theory for embedded surfaces, II

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24 P. B. Kronheimer and T. S. Mrowka (i) Invariants for X o . Let (X,Σ) be an acceptable pair, in the sense of section 2(i). Assume further that π1(X\Σ) has no non-trivial representations in SU(2). We impose also the following condition, which arose in Lemma 3.8: Condition 4.1. The self-intersection number n = Σ·Σ is positive and the genus satisfies the inequality 2g − 2 > 1 2n +2. Now pick an integer k and consider the moduli space Mk(X o ; Rs + ) associated to the manifold with cylindrical end. The dimension of this moduli space is 2d where, as usual, d =4k−3 2 (b+ −b 1 +1) (4.2) by Proposition 3.7. We shall suppose that d is non-negative and that k is strictly positive. Choose closed surfaces in X o representing homology classes u1,...,ud and let V1,...,Vd be zero-sets of sections of the usual line bundles on the corresponding spaces of connections, so that the formal dimension of the cut-down moduli space Mk(X o ; R s +) ∩ V1 ∩ ...∩Vd (4.3) is zero. By a compactly supported perturbation of the metric we can arrange that Mk(X o ; Rs +) and all other moduli spaces Mκ(X o ; Rs )forκ≤kare regular and contain no reducible connections (except for the trivial connection in M0). We can then arrange that the intersection (4.3) is transverse and that the same is true for all partial intersections Mκ(X o ; R s ) ∩ Vi1 ∩ ...∩Vic (4.4) involving the lower moduli spaces and other components R⊂R(Y). Lastly, we ensure that no connections asymptotic to the singular strata in R+ or R− will interfere: using Corollary 3.13 we can arrange that whenever the formal dimension of (4.4) is zero or less, the potentially larger space Mκ(X o ; R) ∩ Vi1 ∩ ...∩Vic (4.5) contains no connections which are not already in (4.4). (Note that Condition 4.1 ensures that the genus g is at least 3). In particular, the spaces (4.5) and (4.4) are both empty if the formal dimension is negative. With all these hypotheses the intersection (4.3) is, first of all, a smooth zerodimensional manifold. From Appendix 1, it is also naturally oriented, and the question of compactness is addressed in the next lemma: Lemma 4.6. If k is sufficiently large that the dimension 2d exceeds 4k, the intersection (4.3) is compact. Proof. Let Ai be a sequence in the intersection (4.3) converging weakly to some A. Let δ be the number of units of charge which is lost in the weak limit due to concentration of curvature at points. Then, by the usual argument [D4] the
Gauge theory for embedded surfaces, II 25 weak limit lives in an intersection such as (4.5), with κ ≤ k − δ and c ≥ d − 2δ. The inequality in the lemma then ensures that c is positive, just as in the usual theory; this in turn means that A is not the trivial flat connection. Consider now the subset (4.4) of the set (4.5) lying over the smooth part of the component of the representation variety. The condition (4.1) on the genus means that Lemma 3.8 and its corollary (3.9) are applicable; this tells us that the formal dimension of Mκ(X o ; R s ) is not more than the dimension of Mk−δ(X o ; R s ± ), which is 2d − δ. So the dimension of (4.4) is at most 2d − 8δ − 2c, which, substituting the previous inequalities, is no more than −4δ. Since this is non-positive, the sets (4.4) and (4.5) coincide: the singular strata do not contribute. Furthermore, since the dimension cannot be negative, δ must be zero, c must be d, κ must be k and R must be R+ or R−. This means that the convergence is strong: the Ai converge to A in the topology which defines the moduli spaces. In particular, R must be R+, this being part of the assertion that the map r to the representation variety is continuous. ⊓⊔ With this standard compactness result in place, we define an invariant q o k (the superscript o indicating the cylindrical end) by the familiar procedure of counting points according to orientation: q o k (u1,...,ud)=# Mk(X o ;R s + )∩V1∩...∩Vd . (4.7) This quantity is independent of the choice of metric and depends only on the homology classes of the surfaces ui; the proof is quite standard, and will be omitted. In any case, as we mentioned at the beginning of this section, we have no need of the full strength of these results. It will be convenient nonetheless to exploit the fact that (4.7) is independent of small compactly-supported perturbations of the metric (as opposed to perturbations which change the geometry of the cross-section of the cylinder) and small isotopies of the embedded surfaces ui, subject to the transversality conditions. This is achieved, as usual, by considering a generic 1-parameter path joining, for example, the two metrics; it was with this purpose in mind that the precise condition (4.1) was selected, as it ensures that the components Mκ(X o ; Rm) will not enter into the weak compactification even of a 1-dimensional parametrized moduli space (see the final clause of (3.8)). We can regard the invariant qo k as defining a polynomial or a multi-linear map q o k : s d (H2(X o )) −→ H0(R+) = Z (4.8) with values in the zero-dimensional homology of R+, which is of course just the integers. In fact, however, as with the invariant qk,l defined in Theorem 2.13, the polynomial q o k factors through H2(X), and can therefore be regarded as a polynomial on [Σ] ⊥ ⊂ H2(X). We shall see below (5.8) that this invariant is nothing new and coincides with the ordinary polynomial invariant qk for the closed manifold, restricted to the homology classes disjoint from Σ.
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Gauge theory for embedded surfaces, II

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