A note on Gorenstein spaces

Abstract

Associated with an augmented differential graded algebra $R=R^{\ge 0}$ is a homotopy invariant $\mathcal{T}(R)$. This is a graded vector space, and if $H^0(R)$ is the ground field and $H^{>N}(R)=0$ then dim$\mathcal{T}(R)=1$ if and only if $H(R)$ is a Poincaré duality algebra. In the case of Sullivan extensions $\wedge W\to \wedge W\otimes \wedge Z\to \wedge Z$ in which dim$H(\wedge Z)<\infty$ we show that $$\mathcal{T}(\wedge W\otimes \wedge Z)=\mathcal{T}(\wedge W)\otimes \mathcal{T}(\wedge Z).$$ This is applied to finite dimensional CW complexes $X$ where the fundamental group $G$ acts nilpotently in the cohomology $H(\tilde{X};\mathbb{Q})$ of the universal covering space. If $H(X;\mathbb{Q})$ is a Poincaré duality algebra and $H(\tilde{X};\mathbb{Q})$ and $H(BG;\mathbb{Q})$ are finite dimensional then they are also Poincaré duality algebras.