A bound on the cohomology of quasiregularly elliptic manifolds

Abstract

We show that a closed, connected and orientable Riemannian manifold of dimension d that admits a quasiregular mapping from R d must have bounded cohomological dimension independent of the distortion of the map. The dimension of the degree l de Rham cohomology of M is bounded above by d l . This is a sharp upper bound that proves the Bonk-Heinonen conjecture [2]. A corollary of this theorem answers an open problem posed by Gromov in 1981 [8]. He asked whether there exists a d-dimensional, simply connected manifold that does not admit a quasiregular map from R d . Our result gives an affirmative answer to this question. EDEN PRYWESThe sphere S 2 satisfies dim H 2 (S 2 ) = 1. By the Künneth formula [3, p. 47], dim H 2 (S 2 × S 2 ) = 2. For 1 ≤ l ≤ d − 2, H l (M #N ) ∼ = H l (M ) ⊕ H l (N ), whenever M and N are smooth manifolds by the Mayer-Vietoris Theorem [3, p. 22]. Therefore dim H 2 (M ) = 2n > 4 2 . So by Theorem 1.1, M is not quasiregularly elliptic.Theorem 1.1 is a generalization of a classical theorem for holomorphic functions in dimension 2. Let M be a Riemann surface, by the uniformization theorem, the universal cover of M is either C, C, or D. If f : C → M is holomorphic, then f lifts to a holomorphic map from C to the universal cover of M . If the universal covering space is D, then Liouville's theorem states that f is constant. This implies that the only compact Riemann surfaces that admit holomorphic mappings are homeomorphic to C and S 1 × S 1 . This proof can be applied to quasiregular mappings in dimension 2 because every quasiregular mapping f = g • φ, where g is holomorphic and φ : C → C is a quasiconformal homeomorphism [15, p. 247].A 1-quasiregular map on C is a holomorphic function. If we study quasiregular ellipticity for K = 1 in higher dimensions, then the results are as restrictive as in the d = 2 case. If M admits a 1-quasiregular mapping from R d , Bonk and Heinonen [2, Proposition 1.4] showed that M must be a quotient of the d-dimensional sphere or torus. For manifolds of dimension 3, Theorem 1.1 is known for each K ≥ 1. Jormakka [13] showed that if M is quasiregularly elliptic then M must be a quotient of S 3 , T 3 , or S 2 × S 1 . One sees that in higher dimensions there are separate results for when K = 1 and when K ≥ 1. In the study of K-quasiregular mappings for d ≥ 4, there are very few conditions on the topology of M that restrict which manifolds can be quasiregularly elliptic, independent of K.A theorem by Varopoulos gives a K-independent result. It states that the polynomial order of growth of the Cayley graph of the fundamental group of a quasiregularly elliptic manifold is bounded by d (see [21, Theorem X.5.1] or [9, Chapter 6]). This result gives a K-independent bound on the size of the fundamental group of the manifold, but does not apply when the fundamental group is small, specifically when the manifold is simply connected.A recent theorem due to Kangasniemi [14] gives a K-independent bound on the cohomology for manifolds that admit uniformly quasiregular self-mappings. H...

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