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Quantitative K-theory, positive scalar curvature, and band width

Hao Guo, Zhizhang Xie, Guoliang Yu

Abstract

We develop two connections between the quantitative framework of operator $K$-theory for geometric $C^*$-algebras and the problem of positive scalar curvature. First, we introduce a quantitative notion of higher index and use it to give a refinement of the well-known obstruction of Rosenberg to positive scalar curvature on closed spin manifolds coming from the higher index of the Dirac operator. We show that on a manifold with uniformly positive scalar curvature, the propagation at which the index of the Dirac operator vanishes is related inversely to the curvature lower bound. Second, we give an approach, using related techniques, to Gromov's band width conjecture, which has been the subject of recent work by Zeidler and Cecchini from a different point of view.

Quantitative K-theory, positive scalar curvature, and band width

Abstract

We develop two connections between the quantitative framework of operator -theory for geometric -algebras and the problem of positive scalar curvature. First, we introduce a quantitative notion of higher index and use it to give a refinement of the well-known obstruction of Rosenberg to positive scalar curvature on closed spin manifolds coming from the higher index of the Dirac operator. We show that on a manifold with uniformly positive scalar curvature, the propagation at which the index of the Dirac operator vanishes is related inversely to the curvature lower bound. Second, we give an approach, using related techniques, to Gromov's band width conjecture, which has been the subject of recent work by Zeidler and Cecchini from a different point of view.

Paper Structure

This paper contains 13 sections, 5 theorems, 147 equations.

Table of Contents

  1. Introduction and motivation
  2. Preliminaries
  3. Notation
  4. Geometric $C^*$-algebras and modules
  5. Quantitative K-theory and higher index
  6. Quantitative K-theory
  7. The quantitative higher index
  8. Higher index
  9. Quantitative higher index
  10. Proof of Theorem \ref{['thm main']}
  11. Gromov's band width conjecture
  12. Preliminaries
  13. Quantitative vanishing of the index

Key Result

Theorem 1.1

Let $M$ be a Riemannian spin manifold with fundamental group $\Gamma$. Let $\kappa$ be the scalar curvature and $D_{\widetilde{M}}$ the lift of the Dirac operator on $M$ to its universal cover $\widetilde{M}$. Fix $0<\varepsilon<\frac{1}{20}$. There exists a constant $\omega_0$ such that for every $ for any $N\geq 7$. The constant $\omega_0$ is independent of the manifold $M$. (See section sec qua

Theorems & Definitions (28)

  • Theorem 1.1
  • Conjecture 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • ...and 18 more