# A generalization of integral Poincaré duality

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 $$\mathbb{Z}$$?

• A lot of work has been done (starting with Dwyer, Greenlees, and Iyengar) and furthered by Greenlees on when a morphism $f \colon R \to k$ of ring spectra is `Gorenstein', see arxiv.org/abs/math/0510247 for example. Taking $R=C^*(X;\mathbb{Q})$ and $f \colon R \to \mathbb{Q}$ the natural map gives you the above definition. Replacing $\mathbb{Q}$ with $\mathbb{Z}$ gives you a reasonable definition. (The theory tends to work best when $k$ is a field however). – Drew Heard May 29 at 17:03
• Thank you for your comment. Are you aware of whether there is a definition of being Gorenstein over $\mathbb{Z}$ without using spectra, or are you saying that there is no such definition available, and the work of Dywer, Greenless, Iyengar addresses this deficiency? – Matt May 29 at 17:22
• I'm not aware of anything else, but that could be more to do with my own ignorance! Looking into the references of D-G-I might help you find other sources (e.g., link.springer.com/article/10.1007/BF02776063) – Drew Heard May 30 at 6:46

Prior to Dwyer-Greenlees-Iyengar, Dwyer and myself (independently) defined Gorenstein conditions for group rings over the sphere $$S[G]$$, i.e., the suspension spectrum of a topological group.

The definition easily extends to the case of a morphism $$R\to k$$.

In the orientable case the definition goes like this:

$$R\to k$$ is said to be Gorenstein of dimension $$d$$ if:

1) $$k$$ is finitely dominated as an $$R$$-module, i.e., $$k$$ is a retract up to homotopy of a finite $$R$$-module (this finiteness condition shouldn't be ignored!), and

2) the derived mapping spectrum $$\hom_R(k,R)$$ is weakly equivalent as an $$R$$-module to $$k[-d] := \Sigma^{-d}k$$.

If one wishes to have an unoriented Gorenstein condition, then it seems to me one needs to replace $$k[-d]$$ by a twisted version of it. In the special case when $$R$$ is an augmented $$k$$-algebra spectrum, we can simply require that the $$k$$-module $$\hom_R(k,R)$$ is equivalent to $$k[-d]$$ as a $$k$$-module (but not necessarily as an $$R$$-module).

If $$R$$ is isn't a $$k$$-algebra, we can fix another $$R$$-module structure on $$k$$, call it $$k^\xi$$, and require that $$\hom_R(k,R)$$ is equivalent to $$k^\xi[-d]$$ as an $$R$$-module.

Returning the the group ring case $$S[G]$$, we have:

Theorem. Assume in addition $$\pi_0(G)$$ is finitely presented. Then following are equivalent:

1) $$S[G] \to S$$ is Gorenstein in dimension $$d$$.

2) $$BG$$ is a (finitely dominated) Poincaré duality space in dimenson $$d$$.

And yes, there is a parallel story over $$\Bbb Z$$, but it is enough to work with dgas instead of ring spectra (for example, the derived $$\hom_{R}(\Bbb Z,R)$$ is a differential graded module for a discrete ring homomorphism $$R\to \Bbb Z$$).