Gauge theory for embedded surfaces, II

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30 P. B. Kronheimer and T. S. Mrowka is transverse to rl and that its image is disjoint from the image of the lower rl ′, for l ′ 0, we can glue X o to (W o ,Σ) to obtain a diffeomorphic copy of (X,Σ). The resulting metric gR on (X,Σ) has a cylindrical neck of length R, and has cone-angle 2π/ν along Σ. Wedenoteby K(X,Σ) the cut-down moduli space by which we hope to calculate the invariant on the left hand side in (5.1): K(X,Σ)=M α k,l ∩ V1 ∩ ...∩Vd. Here the metric gR is used, and the Vi are defined using the same sections as before. Proposition 5.3. For all sufficiently large R, the cut-down moduli space K(X,Σ) is diffeomorphic, by an orientation-preserving diffeomorphism, to the fibre product K(X o ) ×r M α 0,l(W o )={(A, B) | rX(A)=rl(B)}. (5.4) Proposition 5.1 follows from this straight away, because our transversality con- ditions mean that the fibre product is a zero-manifold whose points, counted with signs, add up to the product p0,l × qo k appearing in the proposition. ⊓⊔ Proofof(5.3).For each pair (A, B) in the fibre product, we can find neighbourhoods U1, U2 in Mk(X o ; R s + )andMα 0,l (W o ) respectively so that A is the only connection of K(X o ) lying in U1 and such that the fibre product U1 ×r U2 meets (5.4) alone. The simple gluing map (3.18) then gives an injective map ˜ΦR : U1 ×r U2 −→ B α k,l (X,Σ). If V denotes the intersection V1 ∩ ...∩Vd in B α (X,Σ), then the image of ˜ ΦR meets V transversely in the unique point ˜ ΦR(A, B). By Proposition 3.19, in the version appropriate to the twisted connections, the map ˜ ΦR can be deformed, for all sufficiently large R, toamap ΦR:U1×U2−→ M α k,l (X,Σ). Furthermore, ΦR is C1-close to ˜ ΦR inside Ba k,l by (3.23), so it follows that, for large R, the image of ΦR meets V transversely in one point, just as ˜ ΦR does. In this way, to each point (A, B) in the fibre product (5.4), we obtain a unique point of K(X,Σ)=Mα k,l ∩ V , so defining a map ΨR : K(X o ) ×r M α 0,l (W o ) −→ K (X,Σ). (Note that construction of this map is complicated by the fact that cutting down by V does not ‘commute’ with the gluing map ΦR, though it does commute with the approximation ˜ ΦR; see the remarks following (3.23).)
Gauge theory for embedded surfaces, II 31 All we have to show now is that the map ΨR is surjective when R is large. Let Ri be a sequence approaching infinity, and let Ai be a connection in K(X,Σ) with respect to the metric gRi. After passing to a subsequence, the Ai will converge weakly to a pair (A, B), where for some components R, R ′ in R(Y ). A ∈ Mκ(X o ; R) B ∈ Mι(W o ; R ′ ) Lemma 5.5. In the above situation, the only possibility is that the convergence is actually strong and the limit (A, B) is in K(X o ) ×r M α 0,l (W o ). The desired result follows from this lemma, since Proposition 3.20 then implies that Ai is in the image of the map ΦRi for the appropriate (A, B) and all sufficiently large i, whereas if ΨRi were not surjective we could have arranged that this was not the case. This completes the proof of Proposition 5.3, modulo the lemma. ⊓⊔ Proof of Lemma 5.5. The part of the argument which concerns A is similar to the proof of (4.6). First, the connection A must be non-trivial, by the usual counting. Indeed, the total action of the connections Ai is κ(Ai)=k+2αl − α 2 n which lies between k and k +1/n by (4.14); so there can be at most k points of concentration on X o in the limit, and since we are in the stable range there must be some surfaces ui containing no points of concentration. It follows as usual that A lies in at least one of the Vi and is therefore not the product connection. The formal dimension of the space in which A lives can therefore be assumed to coincide with its actual dimension. If δ is the number of points of concentration in X o , then this space will be Mκ(X o ; R) ∩ Vi1 ∩ ...∩Vic (5.6) for some c ≥ d − 2δ. The action κ is not greater than κ(Ai) − δ, which is less than k +1/n. Since κ must also be C.S.(R) modulo one, and since C.S.(R) isa multiple of 1/n or of a quarter, it follows that κ ≤ k − δ. By Corollary 3.9 then, the dimension of (5.6) is not greater than the dimension of Mk−δ(X o ; R s + ) ∩ Vi1 ∩ ...∩Vic, (5.7) with equality only if R = R+ and κ = k − δ. Here we have made use of our assumption that n is not divisible by 4 to eliminate the possibility that R = R−; see the remark following (3.9). The usual combination of the inequalities shows that the formal dimension of the intersection (5.7) is less than or equal to zero, with equality only if δ =0,inwhichcasec=d. It follows that A lives in K(X o ), as desired.
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Gauge theory for embedded surfaces, II

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