Since the 1974 paper by Peskine and Szpiro, liaison theory via complete intersections, and more generally via Gorenstein varieties, has become a standard tool kit in commutative algebra and algebraic geometry, allowing to compare algebraic features of linked varieties. In this paper we develop a liaison theory via quasi-Gorenstein varieties, a much broader class than Gorenstein varieties: it is not misleading to think that quasi-Gorenstein rings are to Gorenstein rings as manifolds are to spheres. As applications, we derive a connectedness property of quasi-Gorenstein subspace arrangements generalizing previous results by Benedetti and the second author, and we deduce the classical topological Lefschetz duality via the Stanley-Reisner correspondence.