# Seminars & Colloquia Calendar

Geometric Analysis Seminar

## A Bound on the Cohomology of Quasiregularly Elliptic Manifolds

#### Eden Prywes, Princeton University

Location:  Hill 705
Date & time: Tuesday, 05 November 2019 at 2:50PM - 3:50PM

Abstract:  A  classical result gives that if there exists a holomorphic mapping $$f\colon \mathbb C \to M$$, then $$M$$ is homeomorphic to $$S^2$$ or $$S^1\times S^1$$, where $$M$$ is a compact Riemann surface.  I will discuss a generalization of this problem to higher dimensions.   I will show that if $$M$$ is an $$n$$-dimensional, closed, connected, orientable Riemannian manifold that admits a quasiregular mapping from $$\mathbb R^n$$, then the dimension of the degree $$l$$ de Rham cohomology of $$M$$ is bounded above by $$\binom{n}{l}$$.  This is a sharp upper bound that proves a conjecture by Bonk and Heinonen.  A corollary of this theorem answers an open problem posed by Gromov.  He asked whether there exists a simply connected manifold that does not admit a quasiregular map from $$\mathbb R^n$$.  The result gives an affirmative answer to this question.

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