On the intersection ring of graph manifolds

TL;DR

This paper computes the intersection ring of 3D graph manifolds with rational coefficients and characterizes these rings algebraically for tree-structured graphs, revealing obstructions to homology cobordism.

Contribution

It provides the first explicit calculation of intersection rings for graph manifolds and an algebraic characterization for tree graphs, linking topology and algebra.

Findings

Intersection rings are explicitly calculated for graph manifolds.

Algebraic characterization of intersection rings for tree graphs is established.

Obstructions to homology cobordism are identified based on intersection ring properties.

Abstract

We calculate the intersection ring of three-dimensional graph manifolds with rational coefficients and give an algebraic characterization of these rings when the manifold's underlying graph is a tree. We are able to use this characterization to show that the intersection ring obstructs arbitrary three-manifolds from being homology cobordant to certain graph manifolds.