Calabi-Yau categories and Poincare duality spaces

TL;DR

This paper reviews how the singular cochain complex of Poincare duality spaces forms Calabi-Yau categories, connecting algebraic and topological methods in a comprehensive overview.

Contribution

It synthesizes recent developments linking Poincare duality spaces, singular cochain complexes, and Calabi-Yau categories, highlighting the application of Auslander-Reiten theory.

Findings

Poincare duality spaces give rise to Calabi-Yau categories

Application of Auslander-Reiten theory to singular cochain complexes

Integration of rational homotopy and algebraic methods

Abstract

The singular cochain complex of a topological space is a classical object. It is a Differential Graded algebra which has been studied intensively with a range of methods, not least within rational homotopy theory. More recently, the tools of Auslander-Reiten theory have also been applied to the singular cochain complex. One of the highlights is that by these methods, each Poincare duality space gives rise to a Calabi-Yau category. This paper is a review of the theory.