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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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