## 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

### Keywords:

quasiregular ellipticity; de Rham cohomology
Full Text:

### References:

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