Mappings of Bounded Mean Distortion and Cohomology

T 2 are the only closed Riemann surfaces admitting nonconstant conformal mappings from the complex plane. The same rigidity is present in higher dimensions; closed manifolds admitting conformal mappings from Rn are quotients of Sn and Tn, see e.g. [2, Prop. 1.4]. However, if distortion is allowed, simple examples show that the spaces Sk1 × Sk2 × · · · × Skl (k1 + · · ·+ kl = n) receive nonconstant mappings of bounded distortion from R n. A mapping f : M → N between oriented Riemannian n-manifolds is said to be a mapping of bounded distortion, or quasiregular, if f is a Sobolev mapping in W 1,n loc (M ;N) and there exists a constant K ≥ 1 so that |Df(x)| ≤ KJf (x) for almost every x ∈ M, where |Df(x)| is the operator norm of the differentialDf(x) and Jf (x) is the Jacobian determinant of f at x. Mappings we consider are continuous and the Sobolev space W 1,n loc (M ;N) is understood as in [4]. By Reshetnyak’s theorem [17, p. 163], quasiregular mappings are discrete and open, and therefore examples of generalized branched covers. A connected and oriented Riemannian n-manifold receiving a nonconstant (K-)quasiregular mapping from Rn is called (K-)quasiregularly elliptic. By the Uniformization Theorem and the measurable Riemann Mapping Theorem, the only closed quasiregularly elliptic 2-manifolds are S2 and T2. For n = 3, closed quasiregularly elliptic manifolds are by Jormakka’s theorem [10] quotients of S3, S2 × S1, and T3. In higher dimensions such characterizations are not known. In dimension n = 4, a construction of Rickman [19]