Symplectized Torelli mapping tori

Abstract

Examples of aspherical closed symplectic 4-manifolds are presented whose Sullivan minimal models are (1,n)-formal for any n, without being formal. They have as cohomology algebra, signature, canonical class, those of a product of a closed Riemann surface of genus g>1 and an elliptic curve, and the fundamental group, resp. $A_{\infty}$ category defined by their de Rham complex, is isomorphic to that of the product of surfaces up to brackets of order n+1, resp. products of order n+1. Nevertheless, the manifolds do not admit any holomorphic structure. These examples are derived from the fact that the Torelli groups are pro-unipotent. The mapping tori for representatives of suitable classes in the Torelli group are considered, and their product with $S^1$ is symplectized a la Thurston.