On the topology of Symplectic Calabi-Yau 4-manifolds

Abstract

Let M be a 4-manifold with residually finite fundamental group G having b_1(G) > 0. Assume that M carries a symplectic structure with trivial canonical class K = 0 in H^2(M). Using a theorem of Bauer and Li, together with some classical results in 4-manifold topology, we show that for a large class of groups $isdetermineduptohomotopyand,infavorablecircumstances,uptohomeomorphismbyitsfundamentalgroup.ThisisanalogoustowhatwasprovenbyMorgan-Szabointhecaseof{b}_1=0andprovidesfurtherevidencetotheconjecturalclassificationofsymplectic4-manifoldswithK=0$. As a side, we obtain a result that has some independent interest, namely the fact that the fundamental group of a surface bundle over a surface is large, except for the obvious cases.

arXiv comment13 pages. Final version, to appear in the Journal of Topology

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