A bound on the cohomology of quasiregularly elliptic manifolds
A bound on the cohomology of quasiregularly elliptic
manifolds
复制标题
拟正则椭圆流形上同调的一个界
作者：
Eden Prywes
我们证明了一个闭合的、连通的、可定向的维度为 $d$ 的黎曼流形，它允许从 $mathbb R^d$ 进行拟正则映射，它必须具有与映射的畸变无关的有界上同调维度。 $M$ 的 Rham 上同调度 $l$ 的维数以 $inom{d}{l}$ 为界。这是证明 Bonk-Heinonen 猜想的尖锐上限。这个定理的推论回答了格罗莫夫在 1981 年提出的一个开放问题。他询问是否存在一个 $d$ 维的单连通流形，它不承认来自 $mathbb R^d$ 的拟正则映射。我们的结果给了这个问题肯定的答案。
We show that a closed, connected and orientable Riemannian manifold of dimension $d$ that admits a quasiregular mapping from $mathbb R^d$ 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 $inom{d}{l}$. This is a sharp upper bound that proves the Bonk-Heinonen conjecture. A corollary of this theorem answers an open problem posed by Gromov in 1981. He asked whether there exists a $d$-dimensional, simply connected manifold that does not admit a quasiregular map from $mathbb R^d$. Our result gives an affirmative answer to this question.