Variations on Poincar\'e duality for intersection homology

Abstract

Intersection homology with coefficients in a field restores Poincar\'e duality for some spaces with singularities, as pseudomanifolds. But, with coefficients in a ring, the behaviours of manifolds and pseudomanifolds are different. This work is an overview, with proofs and explicit examples, of various possible situations with their properties. We first set up a duality, defined from a cap product, between two intersection cohomologies: the first one arises from a linear dual and the second one from a simplicial blow up. Moreover, from this property, Poincar\'e duality in intersection homology looks like the Poincar\'e-Lefschetz duality of a manifold with boundary. Besides that, an investigation of the coincidence of the two previous cohomologies reveals that the only obstruction to the existence of a Poincar\'e duality is the homology of a well defined complex. This recovers the case of the peripheral sheaf introduced by Goresky and Siegel for compact PL-pseudomanifolds. We also list a series of explicit computations of peripheral intersection cohomology. In particular, we observe that Poincar\'e duality can exist in the presence of torsion in the "critical degree" of the intersection homology of the links of a pseudomanifold.