Nonsmooth Calabi-Yau structures for algebras and coalgebras

TL;DR

This paper establishes a duality between generalized Calabi-Yau dg (co)algebras and symmetric dg (co)algebras, extending classical results without smoothness assumptions, and applies this to characterize Poincaré duality spaces.

Contribution

It introduces a new duality framework for dg (co)algebras and applies it to extend Poincaré duality characterizations to non-simply connected spaces.

Findings

Generalized Calabi-Yau dg (co)algebras are Koszul dual to symmetric dg (co)algebras.

Gorenstein and Frobenius properties are shown to be Koszul dual.

A new characterization of Poincaré duality spaces is provided, extending previous theorems.

Abstract

We show that generalised Calabi-Yau dg (co)algebras are Koszul dual to generalised symmetric dg (co)algebras, without needing to assume any smoothness or properness hypotheses. Similarly, we show that Gorenstein and Frobenius are Koszul dual properties. As an application, we give a new characterisation of Poincar\'e duality spaces, which extends a theorem of F\'elix- Halperin-Thomas to the non-simply connected setting.