Three-Dimensional Manifolds, Skew-Gorenstein Rings and their Cohomology

Abstract

Graded skew-commutative rings occur often in practice. Here are two examples: 1) The cohomology ring of a compact three-dimensional manifold. 2) The cohomology ring of the complement of a hyperplane arrangement (the Orlik-Solomon algebra). We present some applications of the homological theory of these graded skew-commutative rings. In particular we find compact oriented 3-manifolds without boundary for which the Hilbert series of the Yoneda Ext-algebra of the cohomology ring of the fundamental group is an explicit transcendental function. This is only possible for large first Betti numbers of the 3-manifold (bigger than -- or maybe equal to -- 11). We give also examples of 3-manifolds where the Ext-algebra of the cohomology ring of the fundamental group is not finitely generated

arXiv comment25 pages, some minor corrections; deduction of example IV corrected. Appendix added (how to prove that rings are Koszul, using an extension of the Backelin programme BERGMAN), Final version to appear in J.of Commutative Algebra