Sharp cohomological bound for uniformly quasiregularly elliptic manifolds

We show that if a compact, connected, and oriented $n$-manifold $M$ without boundary admits a non-constant non-injective uniformly quasiregular self-map, then the dimension of the real singular cohomology ring $H^*(M;\Bbb{R})$ of $M$ is bounded from above by $2^n$. This is a positive answer to a dyn...

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Published in:American journal of mathematics 2021-08, Vol.143 (4), p.1079-1113
Main Author: Kangasniemi, Ilmari
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Language:English
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Homology
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description We show that if a compact, connected, and oriented $n$-manifold $M$ without boundary admits a non-constant non-injective uniformly quasiregular self-map, then the dimension of the real singular cohomology ring $H^*(M;\Bbb{R})$ of $M$ is bounded from above by $2^n$. This is a positive answer to a dynamical counterpart of the Bonk-Heinonen conjecture on the cohomology bound for quasiregularly elliptic manifolds. The proof is based on an intermediary result that, if $M$ is not a rational homology sphere, then each such uniformly quasiregular self-map on $M$ has a Julia set of positive Lebesgue measure.
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title Sharp cohomological bound for uniformly quasiregularly elliptic manifolds
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