Weak and strong $L^p$-limits of vector fields with finitely many integer singularities in dimension $n$

For every given $p\in [1,+\mathrm{\infty })$ and $n\in \mathbb{N}$ with $n\ge 1$, the authors identify the strong $L^p$-closure $L_{\mathbb{Z}}^p(D)$ of the class of vector fields having finitely many integer topological singularities on a domain $D$ which is either bi-Lipschitz equivalent to the open unit $n$-dimensional cube or to the boundary of the unit $(n+1)$-dimensional cube. Moreover, for every $n\in \mathbb{N}$ with $n\ge 2$ the authors prove that $L_{\mathbb{Z}}^p(D)$ is weakly sequentially closed for every $p\in (1,+\mathrm{\infty })$ whenever $D$ is an open domain in ${\mathbb{R}}^n$ which is bi-Lipschitz equivalent to the open unit cube. As a byproduct of the previous analysis, a useful characterisation of such class of objects is obtained in terms of existence of a (minimal) connection for their singular set.