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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 $d$. …

Dualizable is equivalent to $BG$ being finitely dominated (a retract up to homotopy of a finite cell complex). In Wall's "Finiteness conditions on CW complexes, I" Wall shows that finite domination i …

the intersection $\Delta_\ast \mu \cdot \Delta_\ast \mu$ does not require transversality to define—one may take the Poincaré dual of $\Delta_\ast \mu$, then apply the cup square and then apply Poincaré duality … Suppose now that $M$ is merely an oriented connected Poincaré duality space of dimension $d$, with fundamental class $\mu$. …

1) The answer is yes, at least up to homotopy. This can be found in Wall's paper: Wall, C. T. C., Classification problems in differential topology---IV. Thickenings. Topology, 1966, 5, 73–94. Wall a …

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. … If this conjecture holds, then every finitely presented Poincaré duality group $\pi$ will have the property that $B\pi$ is homotopy finite. …