Poincar\'e dualization and Massey products

TL;DR

This paper explores a construction that extends certain algebraic structures to satisfy Poincaré duality, revealing new properties of Massey products and their behavior under maps between Poincaré duality spaces.

Contribution

It introduces a method to extend cohomologically connected cdgas to satisfy Poincaré duality and analyzes the implications for Massey products and formality in Poincaré duality spaces.

Findings

Quadruple Massey products can pull back trivially under non-zero degree maps.

Non-zero degree maps between formal Poincaré duality spaces may not be formal.

The construction connects Massey products with cyclic A-infinity algebras.

Abstract

We study the rational homotopy theoretic and geometric properties of a construction which extends any cohomologically connected, finite type cdga to one satisfying cohomological Poincar\'e duality. Using this construction we show that non-trivial quadruple Massey products can pull back trivially under non-zero degree maps of Poincar\'e duality spaces, unlike the case of triple Massey products as studied by Taylor. We also show that a non-zero degree map between formal rational Poincar\'e duality spaces need not be formal. Our consideration of Massey products naturally ties in with cyclic $A_\infty$-algebras modelling Poincar\'e duality spaces.