Topological obstructions to continuity of Orlicz-Sobolev mappings of finite distortion

In the paper we investigate continuity of Orlicz-Sobolev mappings $W^{1,P}(M,N)$ of finite distortion between smooth Riemannian $n$-manifolds, $n\ge 2$, under the assumption that the Young function $P$ satisfies the so called divergence condition ${\int }_1^{\mathrm{\infty }}P(t)\mathrm{/}{t}^{n+1}dt=\mathrm{\infty }$. We prove that if the manifolds are oriented, $N$ is compact, and the universal cover of $N$ is not a rational homology sphere, then such mappings are continuous. That includes mappings with $Df\in L^n$ and, more generally, mappings with $Df\in L^n\log^{-1}L$\mathrm{.}Ontheotherhand,ifthespace$W^{1,P}$islargerthan$W^{1,n}$(forexampleif$Df\in L^n\log^{-1}L$),andtheuniversalcoverof$N$ishomeomorphicto$$\mathbb{S}$^n$,$n\neq 4$,orisdiffeomorphicto$$\mathbb{S}$^n$,$n=4$,thenweconstructanexampleofamappingin$W^{1,P}(M,N)$ that has finite distortion and is discontinuous. This demonstrates a new global-to-local phenomenon: both finite distortion and continuity are local properties, but a seemingly local fact that finite distortion implies continuity is a consequence of a global topological property of the target manifold $N$.