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arXiv · 2403.11957

Relative aspherical conjecture and higher codimensional obstruction to positive scalar curvature

Shihang He
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Abstract

Motivated by the solution of the aspherical conjecture up to dimension 5 [CL20][Gro20], we want to study a relative version of the aspherical conjecture. We present a natural condition generalizing the model $X\times\mathbb{T}^k$ to the relative aspherical setting. Such model is closely related to submanifold obstruction of positive scalar curvature (PSC), and would be in similar spirit as [HPS15][CRZ23] in codim 2 case. In codim 3 and 4, we prove results on how 3-manifold obstructs the existence of PSC under our relative aspherical condition, the proof of which relies on a newly introduced geometric quantity called the it spherical width. This could be regarded as a relative version extension of the aspherical conjecture up to dim 5.

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Shihang He. 2024-03-18. Relative aspherical conjecture and higher codimensional obstruction to positive scalar curvature. https://arxiv.org/abs/2403.11957

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