On smooth approximation of integral cycles mod 2

TL;DR

This paper proves that mod 2 integral cycles in Riemannian manifolds can be approximated by smooth submanifolds with nearly the same area, with controlled singularities, extending smooth approximation results to the unoriented case.

Contribution

It establishes an unoriented smooth approximation theorem for mod 2 integral cycles, including estimates on singular sets and conditions for smooth representatives.

Findings

Approximation of mod 2 cycles by smooth submanifolds with controlled area.

Refined estimates on the singular set depending on cycle codimension.

Smooth representatives exist when the homology class admits an embedded smooth cycle.

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.