The Gromov invariants of Ruan-Tian and Taubes

Abstract

C. Taubes has recently defined Gromov invariants for symplectic four-manifolds and related them to the Seiberg-Witten invariants ([T1], [T2]). Independently, Y. Ruan and G. Tian defined symplectic invariants based on ideas of Witten ([RT]). While similar in spirit, these two sets of invariants are quite different in their details.In this note we show that Taubes' Gromov invariants are equal to certain combinations of Ruan-Tian invariants (Theorem 4.5). This link allows us to generalize Taubes' invariants. For each closed symplectic four-manifold, we define a sequence of symplectic invariants Gr δ , δ = 0, 1, 2 . . . . The first of these, Gr 0 , generates Taubes' invariants, which count embedded J-holomorphic curves. The new invariants Gr δ count immersed curves with δ double points.In particular, these results give an independent proof that Taubes' invariants are well-defined. Combined with Taubes' Theorem [T1], they also show that, for symplectic 4-manifolds with b + > 1, some of the Ruan-Tian symplectic invariants agree with the Seiberg-Witten invariants. Gromov InvariantsFix a closed symplectic four-manifold (X, ω). Following the ideas of Gromov and Donaldson, one can define symplectic invariants by introducing an almost complex structure J and counting (with orientation) the number of J-holomorphic curves on X satisfying certain constraints. Unfortunately, technical difficulties make it necessary to modify the straightforward count in order to obtain an invariant. In this section we review the general construction and describe how the technicalities have led to two types of Gromov invariants.Given (X, ω), one can always choose an almost complex structure J tamed by ω, i.e. with ω(Z, JZ) > 0 for all tangent vectors Z. A map f : Σ → X from a topological surface Σ is called J-holomorphic if there is a complex structure j on Σ such thatwhere ∂ J f = 1 2 (df •j−J •df ). The image of such a map is a J-holomorphic curve. Conversely, each immersed J-holomorphic curve is uniquely specified by the equivalence class of a J-holomorphic * partially supported by a M.S.R.I. Postdoctoral Fellowship † partially supported by N.S.F. grant DMS-9626245

A cylindrical reformulation of Heegaard Floer homology

“…There are several differences between our setup and the setup of [7]. We claim the bound (11) for all L . The holomorphic curves in [7] all have source a strip, but this is again essentially irrelevant for his proof.…”

“…In general, @ i counts holomorphic curves with singularity equivalent to i double points, in homology classes A with ind.A/ D 2i C1. (This idea is inspired by the so-called Taubes series described by Ionel-Parker in [11]. The analog for disks in Sym g .…”

A note on the geometry of M 3 and symplectic structures on S 1 × M 3

“…Another information which could be helpful is to understand the Seiberg-Witten invariants of 3-manifolds that fiber over the circle. These have been studied extensively in [2] by Ionel and Parker, in a rather indirect way, under their discussion of Gromov invariants of symplectic manifolds. Since the work of Taubes [4], [5], [6] relate Gromov invariants of symplectic 4-manifolds to the Seiberg-Witten invariants of them, we already have an understanding of the Seiberg-Witten invariants of fibered 3-manifolds.…”

Gromov-Witten Invariants of Symplectic Sums

“…The invariant (1.4) counts (J, ν)-holomorphic maps from connected domains. It is often more natural to work with domains with more components, as Taubes does in [T] (see also [IP1]). Let M χ,n be the space of all compact Riemann surfaces of euler characteristic χ with any number of components, of any possible genus, and with n ordered points distributed in all possible ways.…”