Gauge theory for embedded surfaces, II

More documents

Recommendations

Info

$Math 126 Solution Set 2$

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

32 P. B. Kronheimer and T. S. Mrowka We have now seen that A has action k, which leaves at most κ(Ai) − k for B. This is the quantity 2αl − α 2 n which, by our choice of α, islessthan1/n. We can see now that the convergence is strong on the neck, for if it is not then the quantity lost there would be positive and equal to C.S.(R ′ ) modulo one, which is not consistent with it being less than 1/n. It follows that R = R+ and r(A)=r(B). The action lost at points of concentration in W o is also less than 1/n, soby the same argument as in the proof of (4.13), the only possibility here is that the monopole number l drops due to concentration at points of Σ: any change in the instanton number k would require too much action. It follows that B ∈ M α 0,l ′(W o ,Σ; R+), for some l ′ ≤ l. However, since r(B) =r(A) we cannot have l ′
Gauge theory for embedded surfaces, II 33 though now the fibre product on the left is naturally just K(X o ). The claim is that the map is surjective for all sufficiently large R. Following the line of (5.3) still, we consider a sequence of connections Ai belonging to the set K(X), for a sequence of neck-lengths Ri going to infinity. After passing to a subsequence, we obtain a weak limit (A, B), with A and B being anti-self-dual connections on X o and W o respectively. Lemma 5.9. In the above situation, the only possibility is that A is in K(X o ) and B is flat, in which case the convergence is strong. Proof. The first part of the proof of (5.5) applies, and shows that A is in K(X o ) (we are still assuming that the genus and self-intersection numbers satisfy the inequality laid down in Condition 4.1). The fact that B is flat is an immediate corollary, since the total action cannot be more than k. Because no action is lost, the convergence is strong. ⊓⊔ The surjectivity of ΦR for large R is a straightforward consequence, as before, and since ΦR is now an orientation-preserving diffeomorphism between K(X o ) and K(X), the equality of the two invariants follows. ⊓⊔ Combining Propositions 5.1 and 5.8 directly, we obtain the next result as an immediate corollary. We state it with all the hypotheses in place: Theorem 5.10. Let (X,Σ) be an acceptable pair, with X simply connected. Suppose that the genus g and self-intersection n of Σ satisfy the inequality of (4.1). Suppose in addition that g is odd and that n is not divisible by 4. Letkbe an integer in the stable range and put l = 1 4 (2g − 2). Then we have the relation qk,l(u1,...,ud)=p0,l × qk(ū1,...,ūd). Here the ui are arbitrary classes in H2(X\Σ), andūi denotes the image of ui in H2(X). ⊓⊔ (iii) The vanishing theorem. The final result in this section brings in the inequality between the genus and self-intersection number as it appears in the main theorem (1.1). Theorem 5.11. Suppose that X, Σ, k and l satisfy the conditions in Theorem 5.10, and suppose in addition that the inequality of Theorem 1.1 fails, so (2g − 2)
Page 1 and 2: Gauge theory for embedded surfaces,
Page 31: Gauge theory for embedded surfaces,

Gauge theory for embedded surfaces, II

Create successful ePaper yourself

Delete template?

Save as template?