Rectifiability of flat chains

We prove (without using Federer’s structure theorem) that a finite-mass flat chain over any coefficient group is rectifiable if and only if almost all of its 0-dimensional slices are rectifiable. This implies that every flat chain of finite mass and finite size is rectifiable. It also leads to a simple necessary and sufficient condition on the coefficient group in order for every finite-mass flat chain to be rectifiable. Let G be an abelian group with a norm that makes G a complete metric space. In 1966, Fleming [FL] developed a theory of flat chains with coefficients in G. In case G is the integers or the real numbers, with the usual norm, these flat chains are the integral or real flat chains, respectively, previously introduced by Federer and Fleming [FF]. In case G is the integers modulo p, these are the flat chains modulo p which were later constructed in a different way by Federer [F]. After Fleming’s paper, most research involving flat chains has been limited to integral, real, and modulo p chains. However, recently it has become apparent that more general groups are of considerable interest. For example, other normed groups arise naturally in modelling immiscible fluids and soap bubble clusters [W2], and in proving that various surfaces are areaminimizing [ML]. Perhaps the most profound fact about integral flat chains is the rectifiability theorem: every integral flat chain of finite mass is rectifiable. (The compactness theorem for integral currents is an immediate consequence via *The author was partially funded by NSF Grant DMS-95-04456. 1991 Mathematics Subject Classification. Primary 49Q15; secondary 49Q20.