A mathematical problem solved by Susanna Heikkilä relates to the classification of quasiregularly elliptic 4-manifolds, asking what four-dimensional shapes can be obtained by deforming four-dimensional Euclidean geometry. The article by Heikkilä and Pekka Pankka has been published in the Annals of Mathematics journal.
In 1981, the Russian-French mathematician Misha Gromov, winner of the Abel Prize, asked whether the existence of a quasiregular mapping is guaranteed if the target is simply connected, meaning that its fundamental group is trivial and does not constitute an obstruction. The question remained open until 2019 when Prywes provided a four-dimensional counter-example.
“The main result of my doctoral thesis complements the answer to Gromov’s question, as the result can be used to classify closed simply connected four-dimensional manifolds for which there is a quasiregular mapping from a Euclidean space,” says Postdoctoral Researcher Susanna Heikkilä.
Heikkilä, whose hobbies include knitting, also illustrates the matter through knitted fabric. The knitting was completed for her public examination, where she wanted to describe her research in laypeople’s terms.
The handiwork illustrates the mapping from the plane to a sphere, known as the Alexander map. Heikkilä knitted patches of different colors and assembled them into a chessboard pattern with squares of different colors in the corners. Also needed was a ball with differently colored top and bottom hemispheres. When the chess grid is curved around the ball with the colored corners attached to each other, a gap is left between the squares. This sums up the idea of quasiregular mappings: the gaps can be closed by stretching the fabric.
The path to becoming a mathematician
A career in mathematics was not yet clear in Heikkilä’s mind in general upper secondary school. However, her form teacher, also a teacher of mathematics, recognized her talent and suggested that she should go on to study the subject, which is how Heikkilä ended up in Kumpula Campus at the University of Helsinki.
It was only in her second year of study, in a topology course given by Professor Pankka, that Heikkilä genuinely started to become interested in mathematics. This began the years of collaboration that have culminated in the accomplished article.
In their work, mathematicians focus on thinking. Susanna Heikkilä’s preferred way of working is to hatch ideas with pen and paper. When working with others, a blackboard is a must. © Riitta-Leena Inki
It was clear already at the master’s stage that Heikkilä intended to pursue postgraduate studies, which is why she really put her heart into writing her master’s thesis under Pankka’s supervision. The effort paid off, as the thesis as such was almost ready for use as the first article for a doctoral thesis.
Heikkilä’s master’s thesis, entitled “Restricted cohomology of quasiregularly elliptic manifolds” won the master’s thesis award presented annually by the Academic Association for Mathematics and Natural Sciences—MAL and the Academic Engineers and Architects in Finland TEK trade union. The award draws attention to the study of mathematics, physics and computer science, highlighting the significance of these sciences in Finnish society.
“Having a supportive supervisor and colleagues has made research feel meaningful. I also found my spouse in the same field, although we don’t talk mathematics at home in the evening,” Heikkilä says.
At the beginning of 2025, Heikkilä began working as a postdoctoral researcher at the University of Jyväskylä and is applying for further funding, as she wishes to continue studying the theory of quasiregular mappings and curves.
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Quasiregular elliptic problems
Quasiconformal geometry studies the effect of infinitesimal distortion on the shape of objects. Quasiregular mappings investigate covering questions in quasiconformal geometry. A classic example of such questions is the following result based on the uniformization theorem: the only Riemann surfaces which admit a non-trivial holomorphic mapping from the entire complex plane are a two-dimensional sphere and a two-dimensional torus.
In particular, there are no such mappings for the surfaces of the higher genus. This theorem follows the work of Poincaré and Radón on Riemann surfaces from the early 1900s. Today, this outcome is among the basics in textbooks on Riemann surfaces.
What is particularly interesting is that this result of two-dimensional conformal mappings does not change even if what is examined is quasiregular mappings instead of conformal ones. In higher dimensions, conformal and quasiconformal geometry are radically differentiated. The combination of Martio’s, Rickman’s and Väisälä’s result from 1971 with Zorich’s theorem from 1968 shows that the only simple connected Riemannian manifolds in higher dimensions, for which there is a conformal mapping from a Euclidian space, are the Euclidean space itself and a sphere of equal dimension.
In contrast, quasiregular mappings can be found from a Euclidean space to several different spaces. Such manifolds are called “quasiregularly elliptic.”
In 1981, Gromov asked whether there are closed, simply connected manifolds of higher dimensions that are not quasiregularly elliptic. Fundamentally, Gromov asked whether there is a homological obstruction to quasiregularly elliptic manifolds. The first partial answer to this question was given by Bonk and Heinonen, using a compactness argument based on the de Rham cohomology of differential forms.
Eden Prywes conclusively answered Gromov’s question in 2019, demonstrating that the k-th de Rham cohomology of a closed quasiregular elliptic n-manifold is at most the k-th de Rham cohomology of the n-torus. This result leads to the conclusion that closed manifolds with large de Rham cohomology cannot be quasiregularly elliptic.
“The result we prove with Heikkilä provides an algebraic answer to Gromov’s question. Heuristically, the answer is as follows: for a closed manifold to be quasiregularly elliptic, the intersections of its submanifolds (in homological terms) must be realizable simultaneously in the exterior algebra of a Euclidean space. Formally speaking, this means that there must be an algebra monomorphism from the de Rham cohomology of the closed n-manifold to the exterior algebra of the n-dimensional Euclidean space,” says professor Pankka.
This algebraic result can be used to demonstrate that there are closed manifolds having small cohomology that are not quasiregularly elliptic. Combining this result with the construction of branched covering representations by Piergallini and Zuddas as well as the classification of closed 4-manifolds by Donaldson and Freedman provides a classification for closed simply connected quasiregularly elliptic 4-manifolds: they are precisely the manifolds resulting from up to three connected sums of the product of two 2-spheres or the connected sum of up to three two-dimensional projective spaces with either orientation. This concludes the research initiated by Seppo Rickman on closed simply connected quasiregularly elliptic 4-manifolds.
More information:
Susanna Heikkilä et al, De Rham algebras of closed quasiregularly elliptic manifolds are Euclidean, Annals of Mathematics (2025). DOI: 10.4007/annals.2025.201.2.3
Provided by
University of Helsinki
Citation:
Mathematicians crack 40-year-old problem on quasiregularly elliptic manifolds (2025, March 17)
