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Theses

Contractible 3-manifold 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 mathbb{R} {3}. 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 bb{R} 3.vspace{2mm}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, pi{infty} 1, 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 pi {infty} {1} is homeomorphic to mathbb{R} {3}.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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Jian Wang. Contractible 3-manifold and Positive scalar curvature. Metric Geometry [math.MG]. Université Grenoble Alpes, 2019. English. ⟨NNT : 2019GREAM076⟩. ⟨tel-02953229v2⟩

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