On the topology of symplectic Calabi-Yau 4-manifolds

Abstract

Let M be a 4‐manifold with residually finite fundamental group G having b1(G) > 0. Assume that M carries a symplectic structure with trivial canonical class K=0∈H2(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 M is determined up to homotopy and, in favorable circumstances, up to homeomorphism by its fundamental group. This is analogous to what was proved by Morgan–Szabó in the case of b1=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.

Thompson's group $F$ is not SCY

“…Part of this restriction is reflected in known constraints for their fundamental groups, that we will refer to as SCY groups. In the case of b 1 > 0, these results, for which we refer to [Ba08,Li06b,FV13], corroborate the expectation that such groups are (virtually) poly-Z.…”

Virtual Betti numbers and virtual symplecticity of 4-dimensional mapping tori

“…On the other hand, if X admits a fibration over S 1 , in many cases (see [5]) X is finitely covered by a surface bundle over T 2 . The case of our theorem that is not covered by [4] and [14] is that Y has at least one Seifert fibered JSJ piece.…”

“…The virtual Betti numbers for 4-manifolds which fiber over 2-manifolds were subsequently computed in [4] and [14]. In [4], the virtual Betti numbers for most 4−manifolds which fiber over 3-manifolds were also shown to be ∞.…”

“…Although our Theorem 1.2 considers a different class of fibered 4-manifolds from [4] and [14], there is a significant overlap. When the 4-manifold X is a surface bundle over T 2 , it also admits a fibration over S 1 .…”

A note on collapse, entropy, and vanishing of the Yamabe invariant of symplectic 4-manifolds

“…The reader is directed towards [Li106,Li206,FV13] for further details. In particular, all known symplectic 4-manifolds of zero Kodaira dimension with positive first Betti number are infrasolvmanifolds (see [Hi02] for the definition).…”