  We investigate the homotopy type of the Alexander dual of a simplicial
complex. In general the homotopy type of K does not determine the homotopy type
of its dual K*. Moreover, one can construct for each finitely presented group
G, a simply connected simplicial complex K with fundamental group isomorphic to
G. We study sufficient conditions on K for K* to have the homotopy type of a
sphere. We also extend the simplicial Alexander duality to the context of
reduced lattices.
