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)\$\$ W 1 , P ( M , N ) of finite distortion between smooth Riemannian n-manifolds, \$\$n\backslashge 2\$\$ n ≥ 2 , under the assumption that the Young function P satisfies the so-called divergence condition \$\$\backslashint \_1^\backslashinfty P(t)/t^\n+1\backslash, \backslashhbox \dt=\backslashinfty \$\$ ∫ 1 ∞ P ( t ) / t n + 1 d t = ∞ . 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\backslashin L^n\$\$ D f ∈ L n and, more generally, mappings with \$\$Df\backslashin L^n\backslashlog ^\-1L\$\$ D f ∈ L n log - 1 L . On the other hand, if the space \$\$W^\1,P\$\$ W 1 , P is larger than \$\$W^\1,n\$\$ W 1 , n (for example if \$\$Df\backslashin L^n\backslashlog ^\-1L\$\$ D f ∈ L n log - 1 L ), and the universal cover of N is homeomorphic to \$\$\backslashmathbb \S^n\$\$ S n , \$\$n\backslashne 4\$\$ n ≠ 4 , or is diffeomorphic to \$\$\backslashmathbb \S^n\$\$ S n , \$\$n=4\$\$ n = 4 , then we construct an example of a mapping in \$\$W^\1,P(M,N)\$\$ 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.