In this paper, Felix, Halperin and Thomas define the notion of a Gorenstein space over a field $\mathbb{k}$:

An augmented differential graded algebra $R$ over $\mathbb{k}$ is Gorenstein if $\text{Ext}_R(\mathbb{k},R)$ is concentrated in a single degree and has $\mathbb{k}$-dimension one.

 

$X$ is Gorenstein over $\mathbb{k}$ if the cochain algebra $C^*(X,\mathbb{k})$ is Gorenstein.

This definition is motivated by their subsequent results on this being a generalization of the notion of a Poincaré duality space.

Does there exist a parallel notion of a Gorenstein space over $\mathbb{Z}$ which similarly generalizes Poincaré duality over $\mathbb{Z}$?

EDIT: Alternatively, is it thought that no such generalization is available, so that one needs to use the machinery of symmetric spectra to get such a generalization over $\mathbb{Z}$?

In this paper, Felix, Halperin and Thomas define the notion of a Gorenstein space over a field $\mathbb{k}$:

An augmented differential graded algebra $R$ over $\mathbb{k}$ is Gorenstein if $\text{Ext}_R(\mathbb{k},R)$ is concentrated in a single degree and has $\mathbb{k}$-dimension one.

$X$ is Gorenstein over $\mathbb{k}$ if the cochain algebra $C^*(X,\mathbb{k})$ is Gorenstein.

This definition is motivated by their subsequent results on this being a generalization of the notion of a Poincaré duality space.

Does there exist a parallel notion of a Gorenstein space over $\mathbb{Z}$ which similarly generalizes Poincaré duality over $\mathbb{Z}$?

EDIT: Alternatively, is it thought that no such generalization is available, so that one needs to use the machinery of symmetric spectra to get such a generalization over $\mathbb{Z}$?

In this paper, Felix, Halperin and Thomas define the notion of a Gorenstein space over a field $\mathbb{k}$:

An augmented differential graded algebra $R$ over $\mathbb{k}$ is Gorenstein if $\text{Ext}_R(\mathbb{k},R)$ is concentrated in a single degree and has $\mathbb{k}$-dimension one.

$X$ is Gorenstein over $\mathbb{k}$ if the cochain algebra $C^*(X,\mathbb{k})$ is Gorenstein.

This definition is motivated by their subsequent results on this being a generalization of the notion of a Poincaré duality space.

Does there exist a parallel notion of a Gorenstein space over $\mathbb{Z}$ which similarly generalizes Poincaré duality over $\mathbb{Z}$?

In this paper, Felix, Halperin and Thomas define the notion of a Gorenstein space over a field $\mathbb{k}$:

An augmented differential graded algebra $R$ over $\mathbb{k}$ is Gorenstein if $\text{Ext}_R(\mathbb{k},R)$ is concentrated in a single degree and has $\mathbb{k}$-dimension one.

$X$ is Gorenstein over $\mathbb{k}$ if the cochain algebra $C^*(X,\mathbb{k})$ is Gorenstein.

This definition is motivated by their subsequent results on this being a generalization of the notion of a Poincaré duality space.

Does there exist a parallel notion of a Gorenstein space over $\mathbb{Z}$ which similarly generalizes Poincaré duality over $\mathbb{Z}$?

EDIT: Alternatively, is it thought that no such generalization is available, so that one needs to use the machinery of symmetric spectra to get such a generalization over $\mathbb{Z}$?