Abstract:

In 1981, Gromov asked whether each simply connected closed manifold admits a quasiregular mapping from the Euclidean space of the same dimension. Heuristically, this question asks which properties of a closed manifold prevent us from wrapping that space into a Euclidean space. In two dimensions, the answer is classical and follows from classification of surfaces. In three dimensions this question was solved in the 1980's assuming Thurston's geometrization conjecture. In higher dimensions, this question led to a search for cohomological obstructions for quasiregular mappings, especially for closed 4-manifolds. In 2019, Prywes answered to Gromov's question by obtaining a cohomology bound for closed manifolds admitting a quasiregular map from the Euclidean space. In this talk, I will discuss a result with Susanna Heikkilä (Jyväskylä) which gives an answer for this geometric existence question in terms of embedding of the de Rham cohomology into an exterior algebra of n-dimensional vector space. This yields a classification of closed simply connected 4-manifolds admitting a quasiregular mapping from the Euclidean 4-space.

Bio:

Pekka Pankka is Professor of Mathematics at the University of Helsinki and President of the Finnish Mathematical Society. He received his PhD from the University of Helsinki in 2005. His research lies at the intersection of geometric analysis, quasiconformal and quasiregular mapping theory, metric geometry and geometric topology. He has made important contributions to higher-dimensional quasiregular mapping theory and its connections with geometry and topology. His work has appeared in leading journals, including Annals of Mathematics, Acta Mathematica, and Duke Mathematical Journal.