Some open problems about aspherical closed manifolds. (English) Zbl 1327.57001
Ancona, Vincenzo (ed.) et al., Trends in contemporary mathematics. Selected talks based on the presentations at the INdAM day, June 18, 2014. Cham: Springer (ISBN 978-3-319-05253-3/hbk; 978-3-319-05254-0/ebook). Springer INdAM Series 8, 33-46 (2014).
From the introduction: This article is devoted to aspherical closed manifolds and open conjectures, problems and questions about them. All the problems stated here are very interesting and hard. Any progress towards an answer is welcome and valuable. We hope that a reader may be motivated by this note to study them.
We will address the questions whether an aspherical closed manifold is topologically rigid, whether a finitely presented Poincaré duality group is the fundamental group of an aspherical closed manifold, whether an aspherical closed manifold carries an \(S^1\)-action or a Riemannian metric with positive scalar curvature, and finally state some conjectures about the possible values of \(L^2\)-Betti numbers and \(L^2\)-torsion of the universal covering of an aspherical closed manifold and the homological growth in a tower of finite coverings.
For the entire collection see [Zbl 1297.00035].
We will address the questions whether an aspherical closed manifold is topologically rigid, whether a finitely presented Poincaré duality group is the fundamental group of an aspherical closed manifold, whether an aspherical closed manifold carries an \(S^1\)-action or a Riemannian metric with positive scalar curvature, and finally state some conjectures about the possible values of \(L^2\)-Betti numbers and \(L^2\)-torsion of the universal covering of an aspherical closed manifold and the homological growth in a tower of finite coverings.
For the entire collection see [Zbl 1297.00035].
MSC:
| 57-02 | Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes |
| 57N99 | Topological manifolds |
| 53C24 | Rigidity results |
| 57P10 | Poincaré duality spaces |
| 19-02 | Research exposition (monographs, survey articles) pertaining to \(K\)-theory |
| 20F38 | Other groups related to topology or analysis |
Keywords:
rigidity; Borel conjecture; Singer conjecture; simplicial volume; zero-in-the-spectrum conjecture; random closed manifolds; aspherical closed manifolds; conjectures; Poincaré duality group; fundamental group; \(S^1\)-action; Riemannian metric with positive scalar curvature; \(L^2\)-Betti numbers; \(L^2\) torsion; universal covering; homological growthReferences:
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