Gauge theory for embedded surfaces, II

More documents

Recommendations

Info

$Math 126 Solution Set 2$

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

36 P. B. Kronheimer and T. S. Mrowka (note that S·T is also the geometric intersection number since all intersection points are positive). The formula for the genus then follows. ⊓⊔ As a particular application, let S be a surface with positive self-intersection, and consider perturbing S by moving it along some transverse section of the normal bundle. The result is a nearby surface intersecting S transversely, and by suitable choice of the section we can arrange that all intersection points come with positive signs. We can do this d times, obtaining d copies of S, allof whose mutual intersections are transverse and positive. We can then form their sum in the above sense. The result is a smooth surface which we shall call dS, representing the class d[S]. Its Euler characteristic is χ(dS)=dχ(S) − d(d − 1)S·S. (6.3) The second construction we need is the cancellation of intersection points of opposite sign by means of piping. If x1 and x2 are points of intersection of S and T of opposite sign, choose coordinates in the tubular neighbourhood of S so that, at both points, T coincides with the disk fibre in the normal bundle. Join x1 to x2 by a smooth path γ in S which avoids other intersection points, and let P be the circle bundle of radius ɛ in the normal bundle to S lying over γ. Let D1 and D2 be the disks of radius ɛ in the fibres of the normal bundle to S at x1 and x2, and let T ′ be the piece-wise smooth surface T ′ = T ∪ P \ (D1 ∪ D2). After smoothing the corners of T ′ we have a new oriented surface which no longer intersects S at x1 or x2, and if ɛ is sufficiently small we will have introduced no new intersection points. The genus of T ′ is one greater than the genus of T . The third construction is the modification of an embedded surface by ‘blowing up X at a point of S’. This describes the process of replacing X by ˜ X = X# ¯ CP 2 and replacing S by an internal connect sum ˜S = S#(−E), where E is the sphere generating the homology of ¯ CP 2 and the minus sign denotes that the sum be made so as to reverse the orientation. (The sign is actually unnecessary since there is a diffeomorphism of ˜ X which carries E to −E, but putting the sign in reminds us of the proper transform of S in complex geometry.) The genus of ˜S is the same as that of S, but its self-intersection number is one less since it represents the class [S] − [E] andE·E=−1. We make the following observation concerning this construction: Lemma 6.4. If X is simply connected then, after blowing up at a point of S, the complement ˜ X\ ˜ S is simply connected also. Proof. Since ˜ X is simply connected, the fundamental group of ˜ X\ ˜ S is generated by the conjugacy class of the small loop linking the surface. This loop, however, is contractible in ¯ CP 2 \E. ⊓⊔ The last and most elementary construction we shall need is that of adding a handle to an embedded surface S: the genus of an embedded surface can always
Gauge theory for embedded surfaces, II 37 be increased by 1, without changing the homology class, by forming the internal direct sum S#T with a standard torus T contained in some small ball disjoint from S. (ii) Completion of the proof. Combining theorems (5.10) and (5.11) with (6.1) above, we can deduce the following immediately: Proposition 6.5. Let (X,Σ) be an acceptable pair with X simply connected. Suppose that the self-intersection number n is positive and not divisible by 4, the genus g is odd, and the inequalities 1 2n +2
Page 1 and 2: Gauge theory for embedded surfaces,
Page 35: Gauge theory for embedded surfaces,

Gauge theory for embedded surfaces, II

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?