Poincar\'e dualization and Massey products

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.