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

TL;DR

This paper explores the homological properties of graded skew-commutative rings arising from 3-manifold cohomology and hyperplane arrangements, revealing complex algebraic structures and their implications.

Contribution

It introduces new applications of homological theory to 3-manifold cohomology rings, including explicit transcendental Hilbert series and examples of non-finitely generated Ext-algebras.

Findings

Hilbert series of Ext-algebra can be transcendental for certain 3-manifolds

Existence of 3-manifolds with non-finitely generated Ext-algebras

Large first Betti number (≥11) correlates with complex algebraic structures

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