The remarkable work of Donaldson (e.g., [1]) has revolutionized the ability of topologists to work with smooth 4-manifolds. The past 12 years have led to dramatic advances in our understanding, particularly on issues related to the classification problem for simply connected, compact 4-manifolds. In brief, we have discovered enormous variety among the diffeomorphism types that we can distinguish, and we are left searching for some underlying structure to help us organize these manifolds.
One classical approach to attempting such organization is to try to reduce the question to the theory of complex surfaces (i.e., complex manifolds of complex dimension 2 or real dimension 4). It is easy to show that not every simply connected 4-manifold admits a complex structure -- in fact, a connected sum of two complex surfaces can not even be almost-complex. However, this prompts a question that was open for many years: Is every simply connected, irreducible 4-manifold (other than the 4-sphere) complex? (We call a 4-manifold $M$ irreducible if any connected-sum decomposition of $M$ must involve a homotopy 4-sphere summand. Thus, we rule out connected sums, modulo the 4-dimensional smooth Poincaré conjecture.) The question was finally answered in the negative in 1990 by the author and Mrowka [3]. Subsequent work by numerous mathematicians has shown that irreducible but noncomplex 4-manifolds are quite common. In some sense, they seem to far outnumber the 4-manifolds that admit complex structures. This phenomenon casts some doubt on whether complex surface theory can encompass 4-manifold topology. Of course, one might still hope (by analogy with Thurston's Geometrization Conjecture for 3-manifolds) that any simply-connected 4-manifold can be obtained by simple cut-and-paste techniques from complex surfaces. However, in constrat with torus decompositions of 3-manifolds, there is a shortage of available techniques for decomposing arbitrary 4-manifolds along simple 3-manifolds, presenting difficulties for this approach.
An alternative approach seems promising. Any simply connected complex surface is known to admit a Kähler structure. This can be thought of as a symplectic strucutre (a closed 2-form that is nondegenerate as a bilinear form on each tangent space) that is suitably compatible with the complex structure. We might reject complex structures as being too rigid, and work instead with arbitrary symplectic manifolds. Does this gain us anything? Until recently, few examples of compact symplectic manifolds were known, other than Kähler manifolds. Among simply connected 4-manifolds, it was unknown whether essentially non-Kähler symplectic manifolds could exist. Surprisingly, it turns out that some of the same cut-and-paste techniques that have produced irreducible, noncomplex 4-manifolds can be adpated to construct an enormous variety of non-Kähler symplecti manifolds [2]. For example, many of the simply connected, irreducible, noncomplex 4-manifolds from [3] can be seen to admit symplectic structures. Other examples are not even homotopy equivalent to complex surfaces. In fact, for any finitely presented group $G$ (and any even dimension $\geq 4$) there are families of symplectic manifolds with fundamental group $G$ that are not homotopy equivalent to Kähler manifolds. (In contrast, most groups cannot be realized by Kähler manifolds.) Thus, we obtain much additional freedom in passing from the complex to the symplectic viewpoint on 4-manifolds. There is now evidence to suggest a close relationship between irreducible and symplectic 4-manifolds. Is every simply connected, irreducible 4-manifold $(\not=S^4)$ symplectic? The interplay between 4-manifold and symplectic topology warrants much further study.
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