International Congress of Mathematicians

434 Eleny-Nicoleta IonelThere are other applications of the sum formula (2.3). One such applicationconsidered in [16] begins with the following simple observation. Given any symplecticmanifold X with a codimension 2 symplectic submanifold V, we can write X asa (trivial) symplectic sum X#yFy where Py is the ruled manifold W(NxV ® C)and V is identified with its infinity section. We can then obtain recursive formulasfor the GW invariants of X by moving constraints from one side to the other andapplying the symplectic sum formula.In [15] we used this method to obtain both (a) the Caporaso-Harris formula forthe number of nodal curves in P 2 [2], and (b) the "quasimodular form" expression forthe rational enumerative invariants of the rational elliptic surface [1]. In hindsight,our proof of (a) is essentially the same as that in [2]; using the symplectic sumformula makes the proof considerably shorter and more transparent, but the keyideasare the same. Our proof of (b), however, is completely different from that ofBryan and Leung in [1].We end with another interesting application of the Symplectic Sum Theorem2.1. For each symplectomorphism / of a symplectic manifold X, one can form thesymplectic mapping cylinderX f = X x R x S x /Z (4.1)where the Z action is generated by (x, s, 9) H> (f(x), s + 1,9). In a joint paper [13]with T. H. Parker we regarded Xf as a symplectic sum and computed the Gromovinvariants of the manifolds Xf and of fiber sums of the Xf with other symplecticmanifolds. The result is a large set of interesting non-Kähler symplectic manifoldswith computational ways of distinguishing them. In dimension four this gives asymplectic construction of the 'exotic' elliptic surfaces of Fintushel and Stern [5].In higher dimensions it gives many examples of manifolds which are diffeomorphicbut not 'equivalent' as symplectic manifolds.More precisely, fix a symplectomorphism / of a closed symplectic manifold X,and let /»* denote the induced map on flfe(X;Q). Note that Xf fibers over thetorus T 2 with fiber X. If det (i — /*i) = ±1 then there is a well-defined sectionclass T. Our main result of [13] computes the genus one Gromov invariants of themultiples of this section class. These are the particular GW invariants that, indimension four, CH. Taubes related to the Seiberg-Witten invariants (see [27] and[12]).Theorem 4.1 If det (I—f*i) = ±1, the partial Gromov series of Xf for the sectionclass T is given by the Lefschetz zeta function of f in the variable t = tr-'Gr T (X f ) = Cf(t) IL odd det ( j - tf*k)life even det ( J ~ */**) 'When Xf is a four-manifold, a wealth of examples arise from knots. Associatedto each fibered knot K in S 3 is a Riemann surface S and a monodromydiffeomorphism /#- of S. Taking f = f K gives symplectic 4-manifolds XK of thehomology type of S 2 x T 2 withGr(X K )-ÄK{tT)(i^t T y