On smooth approximation of integral cycles mod 2

Abstract

We prove that every mod 2 integral cycle $T$ in a Riemannian manifold $\mathcal{M}$ can be approximated in flat norm by a cycle which is a smooth submanifold $\Sigma$ of nearly the same area, up to a singular set of codimension 3; in addition, this estimate on the singular set can be refined depending on the codimension of the cycle. Moreover, if the mod 2 homology class $\tau$ admits a smooth embedded representative, then $\Sigma$ can be chosen free of singularities. This article provides the unoriented version of the smooth approximation theorem for integral cycles.