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Theses

Contractible 3-manifolds and Positive scalar curvature

Abstract : The purposes of this thesis is to understand spaces which carry metrics of positive scalar curvature. There are several topological obstructions for a smooth manifold to have a complete metric of positive scalar curvature. Our goal is to find all obstructions for contractible 3-manifolds and closed 4-manifolds. In dimension 3, we are concerned with the question whether a complete contractible 3-manifold of positive scalar curvature is homeomorphic to 3-dimensional Euclidean space. The topological structure of contractible 3-manifolds could be complicated. For example, the Whitehead manifold is a contractible 3-manifold which is not homeomorphic to 3-dimensional Euclidean space. We first prove that the Whitehead manifold does not carry a complete metric of positive scalar curvature. This result can be generalised to the so-called genus one case. Precisely, we show that no contractible genus one 3-manifold admits a complete metric of positive scalar curvature. We then study the fundamental group at infinity and its relationship with the existence of positive scalar curvature metric. The fundamental group at infinity of a manifold is the inverse limit of the fundamental groups of complements of compact subsets. In this thesis, we give a partial answer to the above question. We prove that a complete contractible 3-manifold with positive scalar curvature and trivial fundamental group at infinity is homeomorphic to 3-dimensional Euclidean space. Finally, we study closed aspherical 4-manifolds. Together with minimal surface theory and the geometrisation conjecture, we show that no closed aspherical 4-manifold with non-trivial first Betti number carries a metric of positive scalar curvature.
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Submitted on : Friday, October 2, 2020 - 1:08:15 PM
Last modification on : Wednesday, October 20, 2021 - 3:18:31 AM
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  • HAL Id : tel-02953229, version 1

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Jian Wang. Contractible 3-manifolds and Positive scalar curvature. Differential Geometry [math.DG]. Université Grenoble Alpes, 2019. English. ⟨NNT : ⟩. ⟨tel-02953229v1⟩

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