Gauge theory for embedded surfaces, II

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$Math 126 Solution Set 2$

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

34 P. B. Kronheimer and T. S. Mrowka is strictly less than k. (Recall that l = 1 4 (2g − 2).) Pick such an α, and let let ν be an integer sufficiently large that α is contained in an interval of compactness. With these values of α and ν, we calculate qk,l(u1,...,ud), for some arbitrary set of classes ui, by counting points in the intersection K(X,Σ)=M α k,l ∩ V1 ∩ ...∩Vd. We can suppose that the Riemannian metric gR on (X,Σ) is obtained by attaching X o to W o as before, so that the neck has length R. We assume the usual transversality condition for the multiple intersections (5.6). If the invariant is non-zero, then K(X,Σ) will be non-empty for all values of R. Under this assumption, let Ri be a sequence going to infinity, and let Ai be corresponding connections in K(X,Σ). On passing to a subsequence, there will be weak convergence to a limit (A, B). Consider now the possibilities for the limit connection A on X o . The usual counting argument shows that, because k is in the stable range, the connection A cannot be trivial and must therefore live in a space whose formal dimension (and actual dimension) is non-negative. As usual, if δ is the number of points of concentration of curvature on X o ,thenAlives in a space Mκ(X o ; R) ∩ Vi1 ∩ ...∩Vic with c ≥ d−2δ. The action κ, as a consequence of our choice of α, isstrictly less than k − δ. Since g and n satisfy the hypotheses of (3.9), the dimension of the intersection above is strictly less than the dimension of Mk−δ(X) ∩ Vi1 ∩ ...∩Vic. The dimension of this space however, by the usual count, is −4δ at most, and therefore less than or equal to zero. The strict inequality on the action κ therefore means that the previous intersection, in which A lives, has negative dimension. This contradiction completes the proof. ⊓⊔ Since the arithmetic in the proof above depends on the special value of l at 1 4 (2g − 2), it may be helpful to separate out that part of the argument which is independent of l. The same line of proof as we have just used will prove: Proposition 5.12. Suppose that (X,Σ) satisfies the conditions of Theorem 5.10. Suppose k is in the stable range and let l be arbitrary, but suppose in addition that the following condition holds: there exists α ∈ (0, 1 2 ) such that the action κ satisfies 8κ
Gauge theory for embedded surfaces, II 35 Corollary 5.13. Let (X,Σ) satisfy the conditions of Theorem 5.10 and let k be in the stable range. Then qk,l is zero unless l lies in the range 2(Σ·Σ) − (2g − 2) ≤ 4l ≤ (2g − 2). Proof. The upper and lower bounds on l come from the criterion in the previous proposition, on letting α tend to 0 and 1 2 respectively. ⊓⊔ 6. Proof of the main theorem We shall now give the proof of Theorem 1.1 assuming one result whose proof begins in the following section and occupies the rest of the main body of this paper. Recall that the integer invariant p0,l is associated to a cylindrical end manifold W o , the total space of a line bundle of degree n over a surface of genus g; it is defined whenever n is positive and g is odd and at least 3. The result we shall at present assume is the following: Theorem 6.1. For all positive self-intersection numbers n and all odd genera g greater than 1, the integer invariant p0,l is equal to 2 g . (i) New surfaces from old. We need a few elementary constructions for combining and modifying embedded surfaces. Let S and T be oriented embedded surfaces, intersecting transversely inside an oriented 4-manifold X. There is a canonical way to modify the surfaces in the neighbourhood of each intersection point so as to smooth out the self-intersections: each intersection is replaced by a handle. There are actually two ways of doing this, only one of which will be compatible with the orientation of the two surfaces: the local models for the two possible modifications are illustrated in the way that the singular complex curve xy = 0 in the complex plane C 2 can be deformed either into the smooth curve xy = ɛ or, if the two components are oppositely oriented, into the ‘curve’ x¯y = ɛ (which is not holomorphic). We shall denote by S + T the smooth embedded surface obtained by making the appropriate modification at each point. Lemma 6.2. The self-intersection number of S + T is S·S +2S·T+T·T.Ifall the intersection points have positive sign, then the genus of S + T is g(S + T )=g(S)+g(T)+S·T−1. Proof. The formula for the self-intersection is an immediate consequence of the fact that S +T represents the homology class [S]+[T]. For the second formula, it is easiest first to calculate the Euler characteristic of S +T as χ(S)+χ(T)−2S·T
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Gauge theory for embedded surfaces, II

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