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

TL;DR

This paper explores the algebraic structures of cohomology rings of 3-manifolds and hyperplane arrangement complements, revealing complex properties of their Ext-algebras and applications in homological theory.

Contribution

It introduces new examples of 3-manifolds with complex Ext-algebra structures and applies homological methods to graded skew-commutative rings in topology.

Findings

Existence of 3-manifolds with transcendental Hilbert series of Ext-algebras.

Examples of 3-manifolds with non-finitely generated Ext-algebras.

Identification of conditions on Betti numbers for these properties.

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.