# A note on Gorenstein spaces

## TL;DR

This paper explores the properties of Gorenstein spaces through homotopy invariants associated with differential graded algebras, establishing conditions under which certain cohomology algebras exhibit Poincaré duality.

## Contribution

It introduces a homotopy invariant for augmented differential graded algebras and demonstrates its behavior under Sullivan extensions, linking algebraic properties to topological duality.

## Key findings

- ${
m dim}\, {
m extbf{T}}(R)=1$ iff $H(R)$ is Poincaré duality algebra.
- ${
m extbf{T}}( ext{W} ext{⊗} ext{Z})= {
m extbf{T}}( ext{W}) ext{⊗} {
m extbf{T}}( ext{Z})$ for Sullivan extensions with finite-dimensional cohomology.
- Finite-dimensional Poincaré duality properties transfer from universal covers to base spaces under certain conditions.

## Abstract

Associated with an augmented differential graded algebra $R= R^{\geq 0}$ 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 Poincar\'e duality algebra. In the case of Sullivan extensions $\land W\to \land W\otimes \land Z\to \land Z$ in which dim$\, H(\land Z)<\infty$ we show that $${\mathcal T}(\land W\otimes \land Z)= {\mathcal T}(\land W)\otimes {\mathcal T}(\land Z).$$ This is applied to finite dimensional CW complexes $X$ where the fundamental group $G$ acts nilpotently in the cohomology $H(\widetilde{X};\mathbb Q)$ of the universal covering space. If $H(X;\mathbb Q)$ is a Poincar\'e duality algebra and $H(\widetilde{X};\mathbb Q)$ and $H(BG;\mathbb Q)$ are finite dimensional then they are also Poincar\'e duality algebras.