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Relative aspherical conjecture and higher codimensional obstruction to positive scalar curvature

Shihang He

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.

Relative aspherical conjecture and higher codimensional obstruction to positive scalar curvature

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 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.
Paper Structure (12 sections, 29 theorems, 79 equations)

This paper contains 12 sections, 29 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Enlargeable manifold
  4. Counting the intersection number
  5. Filling estimate and slice and dice
  6. $\mu$-bubble reduction in cubical region
  7. Topological setting for weakly relative aspherical pair
  8. Spherical width and proof of Theorem \ref{['thm1']}
  9. Width of homology class and its estimate at infinity
  10. Reduction to sphere width estimate
  11. The weakly relative aspherical condition
  12. Proof of the Corollaries

Key Result

Theorem 1.1

(CRZ23) Let $Y$ be an orientable connected n-dimensional manifold with $n\le 7, n\ne 5$ and let $X\subset Y$ be a two-sided closed connected incompressible hypersurface which admits no PSC metric. Suppose that one of the following two conditions holds in the case $n\ge6$: (a) $Y$ is almost spin. (b)

Theorems & Definitions (61)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 1.5
  • Remark 1.6
  • Conjecture 1.7
  • Conjecture 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 51 more