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On elliptic and quasiregularly elliptic manifolds

Fedor Manin, Eden Prywes

TL;DR

The paper investigates the relationship between Gromov's ellipticity and quasiregular ellipticity on manifolds, establishing cohomological and group-theoretic obstructions that force closed elliptic and QR-elliptic manifolds to have virtually abelian π1. It shows that, for open manifolds, ellipticity and QR-ellipticity need not coincide and provides explicit constructions. A key methodological feature is constructing scaling limits of differential forms under maps from R^n, yielding injective H^*(M;R) -> Λ^*R^n and topological constraints. The results unify approaches from Gromov–MS, Beurling–Prywes and related works, and they identify several open problems, including whether ellipticity and QR-ellipticity are equivalent on closed manifolds.

Abstract

In his book "Metric structures for Riemannian and non-Riemannian spaces", Gromov defined two properties of Riemannian manifolds, ellipticity and quasiregular ellipticity, and suggested that there may be a connection between the two. Since then, groups of researchers working independently have proved strikingly similar results about these two concepts. We obtain new topological obstructions to the two properties: most notably, we show that closed manifolds of both types must have virtually abelian fundamental group. We also give the first examples of open manifolds which are elliptic but not quasireguarly elliptic and vice versa. Whether there is a direct connection between these properties -- and, in particular, whether they are equivalent for closed manifolds -- remains elusive.

On elliptic and quasiregularly elliptic manifolds

TL;DR

The paper investigates the relationship between Gromov's ellipticity and quasiregular ellipticity on manifolds, establishing cohomological and group-theoretic obstructions that force closed elliptic and QR-elliptic manifolds to have virtually abelian π1. It shows that, for open manifolds, ellipticity and QR-ellipticity need not coincide and provides explicit constructions. A key methodological feature is constructing scaling limits of differential forms under maps from R^n, yielding injective H^*(M;R) -> Λ^*R^n and topological constraints. The results unify approaches from Gromov–MS, Beurling–Prywes and related works, and they identify several open problems, including whether ellipticity and QR-ellipticity are equivalent on closed manifolds.

Abstract

In his book "Metric structures for Riemannian and non-Riemannian spaces", Gromov defined two properties of Riemannian manifolds, ellipticity and quasiregular ellipticity, and suggested that there may be a connection between the two. Since then, groups of researchers working independently have proved strikingly similar results about these two concepts. We obtain new topological obstructions to the two properties: most notably, we show that closed manifolds of both types must have virtually abelian fundamental group. We also give the first examples of open manifolds which are elliptic but not quasireguarly elliptic and vice versa. Whether there is a direct connection between these properties -- and, in particular, whether they are equivalent for closed manifolds -- remains elusive.

Paper Structure

This paper contains 20 sections, 27 theorems, 64 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Definitions and their motivations
  3. Obstructions to ellipticity and quasiregular ellipticity
  4. Construction of elliptic and quasiregularly elliptic manifolds
  5. Open manifolds
  6. Are closed quasiregularly elliptic manifolds elliptic?
  7. Are closed elliptic manifolds quasiregularly elliptic?
  8. Key examples for future work
  9. Proof of the cohomological obstruction
  10. Consequences of the cohomological obstruction
  11. Invariance of ellipticity
  12. Open surfaces
  13. A Gromov-elliptic surface which is conformally hyperbolic
  14. A parabolic surface which is not Gromov-elliptic
  15. Some remarks on surfaces with nontrivial topology
  16. ...and 5 more sections

Key Result

Theorem 1.1

If $M$ is an elliptic or quasiregularly elliptic closed $n$-manifold, then the growth rate of $\pi_1(M)$ is $O(R^n)$. In particular, $\pi_1(M)$ is virtually nilpotent.

Figures (2)

  • Figure 1: The domain of the map $f$. The regions outlined in pink and blue map to the northern and southern hemisphere, respectively. The black curves are preimages of the prime meridian, and each region bounded by the black curves is a homeomorphic preimage of the rest of the sphere. The image is not to scale; it is vertically compressed by a factor of $2\pi$.
  • Figure 2: A schematic illustration of $Q_p$, with the embedding of $S \times [-\frac{1}{2},\frac{1}{2}]$ and several preimages of the corresponding nodule shown.

Theorems & Definitions (47)

  • Theorem 1.1: Varopoulos vsc, Gromov GrMS
  • Theorem 1.2: Berdnikov--Guth--Manin BeGM, Heikkilä--Pankka HP
  • Corollary 1.3
  • Theorem 1.4: Gromov GrMS
  • Theorem 1.5
  • Theorem 1.6
  • Proposition 1.7
  • Theorem 1.8: BeGM
  • Theorem 1.9: Pankka--Rajala PaRa
  • Theorem 1.10
  • ...and 37 more