On the topology of Symplectic Calabi-Yau 4-manifolds

TL;DR

This paper investigates the topology of symplectic Calabi-Yau 4-manifolds with positive first Betti number, showing that their homotopy type is often determined by the fundamental group, supporting classification conjectures.

Contribution

It extends classification results of symplectic Calabi-Yau 4-manifolds to cases with positive first Betti number using Bauer-Li theorem and classical topology.

Findings

Fundamental group often determines the manifold up to homotopy.

Homotopy and sometimes homeomorphism classification for a large class of such manifolds.

Surface bundle groups are large except in obvious cases.

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 $ is determined up to homotopy and, in favorable circumstances, up to homeomorphism by its fundamental group. This is analogous to what was proven by Morgan-Szabo in the case of b_1 = 0 and provides further evidence to the conjectural classification of symplectic 4-manifolds with K = 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.