Pullback of closed forms by low regularity maps to manifolds, and applications

For maps f in the Sobolev space , with \(\mathscr {N}\) $\mathcal{N}$ a closed manifold, Bethuel, Coron, Demengel, and Hélein highlighted the importance, in approximation problems, of the pullbacks \(f^*\omega \) $f^{\ast }\omega$ of smooth closed k-forms \(\omega \) $\omega$ on \({\mathscr {N}}\) $\mathcal{N}$ . When \({\mathscr {N}}\) $\mathcal{N}$ is a sphere-like manifold and \(k\le p<k+1\) $k\le p<k+1$ , they proved that a \(W^{1,p}\) $W^{1,p}$ map to \({\mathscr {N}}\) $\mathcal{N}$ can be strongly approximated with smooth maps to \({\mathscr {N}}\) $\mathcal{N}$ if and only if all its corresponding pullbacks are closed currents. We extend this result to \(W^{s,p}\) $W^{s,p}$ maps, with \(0<s<1\) $0<s<1$ . In the process, we adapt the Brezis–Nirenberg theory of homotopical invariants to VMO maps on metric measure spaces, establish the existence and some main properties of integral invariants for VMO maps on Lipschitz manifolds, prove the existence of distributional pullbacks by fractional Sobolev maps and obtain some of their properties, including various slicing formulas, and characterize the closure of smooth maps in terms of restrictions on generic skeletons.