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I have a couple of questions about (relative) low-dimensional Poincaré-Duality spaces, also known as Poincaré complexes.

From Eckmann-Müller and Eckmann-Linnell we know that a CW complex is a Poincaré 2-complex if and only if it is homotopy equivalent to a closed surface of genus $\ge0$.

Question Is there a modern treatment of this result? Has it been improved since?

I'm interested in the following possible improvements/variations:

  1. The generalization to relative 2-complexes and compact surfaces.
  2. Related statements that might be easier to prove.

Re 2: I'm thinking of having possibly stronger, but still purely homotopy-theoretic conditions that ensure that a homotopy type is realizable as a compact surface. For instance, instead of having a hypothesis on cohomology/homology, does it help if we instead ask for spectrum-level data? It could also be data from parametrized homotopy theory, if that helps.

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    $\begingroup$ I believe the reference for the relative statement is Bieri—Eckmann, Relative homology and Poincare duality for group pairs, 1978. $\endgroup$
    – HJRW
    Commented Mar 29, 2018 at 7:42
  • $\begingroup$ Thanks @HJRW, for bringing that up. It predates the papers on the absolute case (from 1980 and 1983), but it states (p. 314): “In fact it can be proved that the absolute PD²-conjecture implies the relative one.” This is then sketched for one-relator groups. I think this amplifies my question: was this stuff ever revisited later? (And perhaps from a more geometric/homotopical perspective?) $\endgroup$
    – Ulrik Buchholtz
    Commented Mar 29, 2018 at 9:35
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    $\begingroup$ It's true that the relative case follows easily from the absolute case. (Double to obtain an absolute PD_2 group, which is then a surface, and use the fact that any splitting of a surface over a cyclic subgroup is induced by a simple closed curve.) To my mind, the main geometric thread that follows from this is the work on the Gromov (and Bowditch) boundaries of (relatively) hyperbolic groups. The Convergence Group Theorem of Tukia, Casson--Jungreis and Gabai asserts that a hyperbolic group with circle boundary is commensurable to a surface group. Tukia also did the relative case. (cont'd) $\endgroup$
    – HJRW
    Commented Mar 29, 2018 at 13:00
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    $\begingroup$ ... By the Bestvina--Mess theorem (which relates the Cech cohomology of the boundary to the group cohomology), you can deduce the classification of PD_2 groups in the hyperbolic case. The 3-dimensional analogue of the Convergence Group Theorem is the Cannon Conjecture, which is probably the most important component of the question of whether all PD_3 groups are 3-manifold groups. $\endgroup$
    – HJRW
    Commented Mar 29, 2018 at 13:03

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