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A bound on the cohomology of quasiregularly elliptic manifolds. (English) Zbl 1451.30045

The author presents a theorem which proves the Bonk-Heinonen conjecture on dimensions of the cohomology for quasiregular manifolds and answers an open problem posed by Gromov in 1981. For a closed, connected and orientable Riemannian manifold \(M\) of dimension \(d\), a \(K\)-quasiregular mapping, \(K\geq1\), is a continuous mapping \(f:\mathbb R^d\to M\) such that \(f\in W^{1,d}_{\text{loc}}(\mathbb R^d,M)\) and the differential, \(Df:T\mathbb R^d\to TM\), satisfies \[\|Df(x)\|^d\leq KJ_f(x),\;\;\;J_f=\text{det}(Df),\] for almost every \(x\in\mathbb R^d\). If \(M\) admits such a nonconstant quasiregular mapping, then \(M\) is quasiregularly elliptic. The main result is given in the following theorem.
Theorem 1.1. Let \(M\) be a closed, connected and orientable Riemannian manifold of dimension \(d\). If \(M\) admits a nonconstant quasiregular mapping from \(\mathbb R^d\), then \(\text{dim}\,H^l(M)\leq\binom dl\) for \(0\leq l\leq d\), where \(H^l(M)\) is the de Rham cohomology of \(M\) of degree \(l\).
Denote by \(S^2\) the 2-sphere. Theorem 1.1 implies the next corollary.
Corollary 1.2. The simply connected manifold \(M=\#^n(S^2\times S^2)\), the connected sum of \(n\) copies of \(S^2\times S^2\), is not quasiregularly elliptic for \(n\geq4\).

MSC:

30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
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