Graph mappings and Poincaré duality

We give a new geometric description for a compact, oriented, pseduo-manifold X of the Poincaré duality map from the integral cohomology of X to the integral homology of X. Our construction takes a multi-valued Lipschitz map on X with values in a sphere S ⁿ to its geometric-measure-theoretic graph in X × S ⁿ and then to the slice of this graph as an integral cycle on X. This construction is compatible with analogous constructions on algebraic cocycles on projective varieties employed by the authors and others.