Optimal smooth approximation of integral cycles

Abstract

In this article we prove that each integral cycle $T$ in an oriented Riemannian manifold $\mathcal{M}$ can be approximated in flat norm by an integral cycle in the same homology class which is a smooth submanifold $\hat{I}\pounds$ of nearly the same area, up to a singular set of codimension 5. Moreover, if the homology class $\ddot{I}„$ is representable by a smooth submanifold, then $\hat{I}\pounds$ can be chosen free of singularities.