$\mathbb{Z}_k$-smooth approximation of mod $k$ integral cycles

Abstract

We prove that every mod $k$ integral cycle $T$ of sufficiently high codimension in an oriented Riemannian manifold $\mathcal{M}$ can be approximated in flat norm by a cycle which is a $\mathbb{Z}_k$-smooth submanifold $Σ$ of nearly the same area, up to a singular set of codimension 5. If in addition $T$ belongs to a mod $k$ homology class admitting an embedded $\mathbb{Z}_k$-smooth manifold representative, then $Σ$ can be taken fully regular, up to the codimension one stratum of classical singularities. This extends the smooth approximation theory to cycles with coefficients in $\mathbb{Z}/k\mathbb{Z}$ in stable regimes. Moreover, we provide examples showing the impossibility of locally approximating minimizing cycles by smooth or $\mathbb{Z}_k$-smooth submanifolds, even in homology classes with smooth or $\mathbb{Z}_k$-smooth representatives.