A note on Gorenstein spaces

Abstract

Associated with an augmented differential graded algebra R=R≥0 is a homotopy invariant T(R). This is a graded vector space, and if H0(R) is the ground field and H>N(R)=0 then dimT(R)=1 if and only if H(R) is a Poincaré duality algebra. In the case of Sullivan extensions ∧W→∧W⊗∧Z→∧Z in which dimH(∧Z)<∞ we show thatT(∧W⊗∧Z)=T(∧W)⊗T(∧Z). This is applied to finite dimensional CW complexes X where the fundamental group G acts nilpotently in the cohomology H(X˜;Q) of the universal covering space. If H(X;Q) is a Poincaré duality algebra and H(X˜;Q) and H(BG;Q) are finite dimensional then they are also Poincaré duality algebras.