The structure theory of nilspaces. II: Representation as nilmanifolds. (English) Zbl 1440.37017
The present paper is the second one in a series of three papers devoted to the structure theory of cubespaces and nilspaces. Its starting point is the following structural type statement (exposed in Section 1):
Theorem S. Let \(X =(X,C^n(X))\) be a compact ergodic nilspace of degree \(s\) with the properties:
i) \(X\) is locally connected and of finite Lebesgue covering dimension;
ii) all the spaces \(C^n(X)\) are connected.
Then, \(X\) is isomorphic to a nilmanifold \(G/\Gamma\). More precisely, there exists a filtered connected Lie group \(G_*\), a discrete co-compact subgroup \(\Gamma\) of \(G\), and a homeomorphism \(\phi:X\leftrightarrow G/\Gamma\) that identifies the cubes \(C^k(X)\) with the Host-Kra cubes \(HK^k(G_*)/\Gamma\).
Section 2 is essentially devoted to different version of Theorem S. Then, Sections 3 and 4 reduce Theorem S to a cohomological-type statement. Finally, Section 5 gives a proof of this statement via cocycle theory.
For Part I, see [the authors, J. Anal. Math. 140, No. 1, 299–369 (2020; Zbl 1442.37020)].
Theorem S. Let \(X =(X,C^n(X))\) be a compact ergodic nilspace of degree \(s\) with the properties:
i) \(X\) is locally connected and of finite Lebesgue covering dimension;
ii) all the spaces \(C^n(X)\) are connected.
Then, \(X\) is isomorphic to a nilmanifold \(G/\Gamma\). More precisely, there exists a filtered connected Lie group \(G_*\), a discrete co-compact subgroup \(\Gamma\) of \(G\), and a homeomorphism \(\phi:X\leftrightarrow G/\Gamma\) that identifies the cubes \(C^k(X)\) with the Host-Kra cubes \(HK^k(G_*)/\Gamma\).
Section 2 is essentially devoted to different version of Theorem S. Then, Sections 3 and 4 reduce Theorem S to a cohomological-type statement. Finally, Section 5 gives a proof of this statement via cocycle theory.
For Part I, see [the authors, J. Anal. Math. 140, No. 1, 299–369 (2020; Zbl 1442.37020)].
Reviewer: Mihai Turinici (Iaşi)
MSC:
| 37B02 | Dynamics in general topological spaces |
| 37B05 | Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.) |
| 37C85 | Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\) |
| 22E05 | Local Lie groups |
| 53C15 | General geometric structures on manifolds (almost complex, almost product structures, etc.) |
| 57S20 | Noncompact Lie groups of transformations |
Citations:
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