Here are some definitions:


A space is *homotopy finite* if it is homotopy equivalent to a finite CW complex.
A space *finitely dominated* if it is a retract of a homotopy finite space.


A space $X$ is a *Poincaré duality space* of dimension $d$ if there exists a pair
$$
({\mathscr L},[X])
$$
consisting of a rank one local system $\mathscr L$ on $X$ (i.e., a local coefficient system on $X$ which is locally isomorphic to $\Bbb Z$) and $[X] \in H\_d(X;{\mathscr L})$ is a twisted homology class such that the cap product homomorphism
$$
\cap [X]: H^\ast(X;{\mathscr E}) \to H\_{d-\ast}(X;{\mathscr E} \otimes {\mathscr L})
$$
is an isomorphism in all degrees, where $\mathscr E$ runs over all local systems on $X$.


My question is this:


**Question:** *Are there finitely dominated Poincaré duality spaces which are **not** homotopy finite?*


(Note: I am a bit embarrassed about not knowing the answer to this question.)


**Remarks:**


(1). If $X$ is a finitely dominated space with finitely presented fundamental group $\pi$, then Wall's finiteness obstruction $w(X) \in K\_0({\Bbb Z}[\pi])$ is defined. It is known that $X$ is homotopy finite if and only if $w(X)$ lies in the summand
${\Bbb Z} \cong K\_0({\Bbb Z}) \subset K\_0({\Bbb Z}[\pi])$.


(2). When $X= B\pi$ where $\pi$ is a discrete, finitely presented group and $X$ satisfies Poincaré duality, then $\pi$ is called a *Poincaré duality group.* In this case it automatically follows that $X$ is finitely dominated and $\pi$ is torsion free. It remains an open question as to whether $X$ is homotopy finite. However, it is known by work of Ian Leary that $w(X)$ is always a $2$-torsion element. Consequently, if $K\_0({\Bbb Z}[\pi])$ contains no 2-torsion, it follows that $X$ is homotopy finite.


(3). It has been conjectured that for any torsion free group $\pi$, the class group $K\_0(\Bbb Z[\pi])$ is torsion-free. If this conjecture holds, then every finitely presented Poincaré duality group $\pi$ will have the property that $B\pi$ is homotopy finite.



Finite domination and Poincaré duality spaces
Corollary 5.4.2 of Wall's article `Poincaré complexes I', *Ann. Math.* **86** (1967) 213-245 gives examples of 4-dimensional Poincaré complexes $X$ with fundamental group of prime order $p\geq 23$ for which the Wall finiteness obstruction $\chi(X)$ is non-zero.


Incidentally, Theorem 1.3 of the same article is very close to the result that you attributed to me, except that I put in extra hypotheses that imply that (in Wall's notation) $\sigma(X)^\*=\sigma(X)$. The thing I thought of as new in my article was considering PD groups over other rings such as $\mathbb{Q}$.


I think that Wall's $\sigma(X)$ is your $w(X)$ and Wall's $\chi(X)$ is the image of your $w(X)$ in the quotient $K\_0(\mathbb{Z}[\pi])/K\_0(\mathbb{Z})$.


