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It is conjectured that for a discrete, finitely presented group $G$ such that $BG$ satisfies Poincaré duality, there actually exists a closed manifold $M$ which is homotopy equivalent to $BG$.

This is somehow pointing in the opposite direction as Borel's conjecture, which implies that the homeomorphism type of such a manifold $M$ is uniquely determined.

Who conjectured this first? Is it also due to Borel, or was it Wall, or somebody else?

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    $\begingroup$ One way to state the relation with Borel's conjecture is: consider the map $\pi_1$ from homeomorphism classes of closed aspherical manifolds to f.p. PD groups. Borel says the map is injective, and this conjecture says it's surjective. $\endgroup$
    – Kevin Casto
    Commented Mar 6, 2017 at 22:44
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    $\begingroup$ Sure. But was it also Borel who conjectured it, or was it somebody else? $\endgroup$
    – Jens Reinhold
    Commented Mar 6, 2017 at 23:43

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The reference for the first appearance of the conjecture (still without the condition that the PD group has to be a priori finitely presented) seems to be http://www.worldcat.org/title/homological-group-theory-proceedings-of-a-symposium-held-at-durham-in-september-1977-on-homological-and-combinatorial-techniques-in-group-theory/oclc/6022486 from 1977.

You find a survey on the state of the subject around 2000 in Davis: "Poincaré duality groups".

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    $\begingroup$ Highlights from Davis: (1) You can't ask for a smooth, or even PL manifold; (2) Generalizing the Borel conjecture to manifolds with boundary leads to the assembly conjectures, from which the PD conjecture follows. Some versions produce a manifold model. But some people don't expect that and expect a weaker assembly conjecture which merely produces an ANR homology manifold model, sometimes with nontrivial Quinn invariant, obstructing a manifold structure. $\endgroup$
    – Ben Wieland
    Commented Mar 7, 2017 at 4:14

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