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arXiv · 2605.28622

Topological singular set of manifold-valued maps weakly approximable by smooth maps

Giacomo Canevari·Giandomenico Orlandi
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Abstract

Given a positive integer $p$, we consider $W^{1,p}$-maps from a Euclidean domain of dimension $p+1$ into a closed Riemannian manifold $\mathcal{N}$. The target manifold is required to satisfy suitable topological conditions; in particular, the action of $\pi_1(\mathcal{N})$ over the $\pi_p(\mathcal{N})$ must be trivial. However, we do not assume that $\mathcal{N}$ is $(p-1)$-connected. Using tools from geometric measure theory -- namely, flat chains with coefficients in~$\pi_p(\mathcal{N})$ -- we associate to each map $u$ in the weak sequential closure of smooth maps an object that captures its point singularities. The vanishing of this object characterizes local strong approximability by smooth maps.

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Giacomo Canevari, Giandomenico Orlandi. 2026-05-27. Topological singular set of manifold-valued maps weakly approximable by smooth maps. https://arxiv.org/abs/2605.28622

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