# Finite domination and Poincaré duality spaces

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.

• I don't know the answer, but here is a suggestion : if $Y\to X$ is a finite covering space and $Y$ is PD, then so is $X$. So one could try to look for a (simply-connected ?) PD space $Y$ with a finite group action that realizes some nontrivial class in $\widetilde K_0(\mathbb Z[\pi])$. Maybe some manifold with an exotic action ? Commented Oct 5, 2022 at 8:15

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 $$K_0(\mathbb{Z}[\pi])/K_0(\mathbb{Z})$$.