A note on the homotopy type of the Alexander dual

Abstract

We investigate the homotopy type of the Alexander dual of a simplicial complex. It is known that in general the homotopy type of K does not determine the homotopy type of its dual K∗ . We construct for each finitely presented group G, a simply connected simplicial complex K such that π1(K∗ ) = G and study sufficient conditions on K for K∗ to have the homotopy type of a sphere. We extend the simplicial Alexander duality to the more general context of reduced lattices and relate this construction with Bier spheres using deleted joins of lattices. Finally we introduce an alternative dual, in the context of reduced lattices, with the same homotopy type as the Alexander dual but smaller and simpler to compute.