A note on the homotopy type of the Alexander dual
  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. 
Elias Gabriel Minian and Jorge Tomas Rodriguez
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