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.