nLab
Poincare duality

Properties

From ALGTOP-L, Oct 5, 2010.

• Jim Stasheff:

  Any one have a reference for obstructions which detect whether a space whose cohomology has Poincare duality is actually a manifold? thanks

• John Klein:

  Ranicki’s total surgery obstruction does it in the top case, in surgery dimensions.

  • The total surgery obstruction. Algebraic topology, Aarhus 1978 (Proc. Sympos., Univ. Aarhus, Aarhus, 1978), pp. 275–316, Lecture Notes in Math., 763, Springer, Berlin, 1979.

  You can get this article from Andrew’s home page.

• Nathanien Rounds:

  I believe this question is answered (simply connected case, over Q) in Dennis Sullivan’s paper Infinitesmal Computations in Topology (Theorem 13.2). The answer, as I understand it, is that outside dimension 4k any graded commutative algebra over Q wtih first betti number 0 satisfying Poincare Duality can be realized as the cohomoloyg ring of a manifold. In dimension 4k there is an obstruction related to the signature which is given in that paper. There won’t in general be a unique manifold corresponding to the ring; for example one can choose different rational Pontragin classes and change the homemorphism type but not the cohomology.

  The general answer (not simply connected, over Z) is given by Ranicki’s total surgery obstruction, as John Klein has already pointed out. One way to interpret this obstruction geometrically is that the Poincare duality map is always “local” but it need not have a “local” inverse, and the lack of a local inverse is an obstruction to having a anifold structure. See for example McCrory’s paper “A Characterization of Homology Manifolds” and also my thesis, which if you’re interested is here:

  www.math.purdue.edu/~nrounds

Relation to Thom isomorphism

The Poincaré dual of a submanifold can be identified with the Thom class on its normal bundle

(…)

Generalizations

The six operations of Grothendieck and Grothendieck duality are designed to ensure a generalized and relative versions of Poincaré duality and related phenomena in the setups like sheaf and topos theory, algebraic geometry. See there for references.

Another generalization, for singular spaces, is with help of stratifications and via intersection cohomology.

References

For Hochschild cohomology

Poincaré duality on Hochschild (co)homology

• M. Van den Bergh, A relation between Hochschild homology and cohomology for Gorenstein rings . Proc. Amer. Math. Soc. 126 (1998), 1345–1348; (JSTOR)

  Correction: Proc. Amer. Math. Soc. 130 (2002), 2809–2810.

with more on that in

• U. Krähmer, Poincaré duality in Hochschild cohomology (pdf)

That this Poincare duality takes the Connes coboundary operator? to the BV operator is shown in

• Victor Ginzburg, Calabi-Yau Algebras, arXiv/math.AG/0612139