We prove that every polyharmonic map u ∈ W m,2 (B n , S N −1) is smooth in the critical dimension n = 2m. Moreover, in every dimension n, a weak limit u ∈ W m,2 (B n , S N −1) of a sequence of polyharmonic maps u j ∈ W m,2 (B n , S N −1) is also polyharmonic. The proofs are based on the equivalence of the polyharmonic map equations with a system of lower order conservation laws in divergence-like form. The proof of regularity in dimension 2m uses estimates by Riesz potentials and Sobolev inequalities; it can be generalized to a wide class of nonlinear elliptic systems of order 2m.

Abstract. In the paper we investigate the degree and the homotopy theory of Orlicz-Sobolev mappings W 1,P (M, N ) between manifolds, where the Young function P satisfies a divergence condition and forms a slightly larger space than W 1,n , n = dim M . In particular, we prove that if M and N are compact oriented manifolds without boundary and dim M = dim N = n, then the degree is well defined in W 1,P (M, N ) if and only if the universal cover of N is not a rational homology sphere, and in the case n = 4, if and only if N is not homeomorphic to S 4 .

Abstract. We construct an almost everywhere approximately differentiable, orientation and measure preserving homeomorphism of a unit n-dimensional cube onto itself, whose Jacobian is equal to −1 a.e. Moreover we prove that our homeomorphism can be uniformly approximated by orientation and measure preserving diffeomorphisms.

We prove the following dichotomy: if n = 2, 3 and f ∈ C 1 (S n+1 , S n ) is not homotopic to a constant map, then there is an open set Ω ⊂ S n+1 such that rank df = n on Ω and f (Ω) is dense in S n , while for any n ≥ 4, there is a map f ∈ C 1 (S n+1 , S n ) that is not homotopic to a constant map and such that rank df < n everywhere. The result in the case n ≥ 4 answers a question of Larry Guth.