The generalized Kummer construction and the cohomology rings of $G_2$-manifolds

TL;DR

This paper uses intersection theory to compute the cohomology rings of G_2-manifolds constructed via the generalized Kummer method, providing explicit generators and product structures for at least one example.

Contribution

It introduces a method to explicitly calculate the cohomology rings of G_2-manifolds obtained through the generalized Kummer construction, including generators and product structures.

Findings

Computed the rational cohomology ring for a G_2-manifold example

Identified generators of the cohomology ring

Described the product structure within the cohomology ring

Abstract

We demonstrate how by using the intersection theory to calculate the cohomology of $G_2$-manifolds constructed by using the generalized Kummer construction. For one example we find the generators of the rational cohomology ring and describe the product structure.