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Relative higher index theory on quotients of Roe algebras and positive scalar curvature at infinity

Liang Guo, Qin Wang, Chen Zhang

TL;DR

This work develops a relative higher index theory for quotients of Roe algebras by ghostly ideals to study obstructions to uniformly positive scalar curvature at infinity on non-compact manifolds. By formulating relative (maximal) coarse Baum–Connes and Novikov conjectures and introducing relative localization algebras, it provides a refined framework that localizes index data near infinity via the quotient $K$-theory. The authors prove key results for spaces admitting relative fibred coarse embeddings (RFCE), showing the relative assembly maps are isomorphisms and deriving PSC obstructions that sharpen Gromov–Lawson-type results. They further derive global consequences from relative data, including maximal BC for products of RFCE spaces and demonstrations of subset-sensitivity in the relative conjectures. Overall, the approach yields both conceptual insights and computable tools for PSC obstructions and coarse geometric conjectures in the non-compact setting.

Abstract

In this paper, we employ quotients of Roe algebras as index containers for elliptic differential operators to study the existence problem of Riemannian metrics with positive scalar curvature on non-compact complete Riemannian manifolds. The non-vanishing of such an index locates the precise direction at infinity of the obstructions to positive scalar curvature, and may be viewed as a refinement of the positive scalar curvature problem. To achieve this, we formulate the relative coarse Baum-Connes conjecture and the relative coarse Novikov conjecture, together with their maximal versions, for general metric spaces as a program to compute the $K$-theory of the quotients of the Roe algebras relative to specific ideals. We show that if the metric space admits a relative fibred coarse embedding into Hilbert space or an $\ell^p$-space, certain cases of these conjectures can be verified, which yield obstructions to the existence of uniformly positive scalar curvature metrics in specified directions at infinity. As an application, we prove that the maximal coarse Baum-Connes conjecture holds for finite products of certain expander graphs that fail to admit fibred coarse embeddings into Hilbert space.

Relative higher index theory on quotients of Roe algebras and positive scalar curvature at infinity

TL;DR

This work develops a relative higher index theory for quotients of Roe algebras by ghostly ideals to study obstructions to uniformly positive scalar curvature at infinity on non-compact manifolds. By formulating relative (maximal) coarse Baum–Connes and Novikov conjectures and introducing relative localization algebras, it provides a refined framework that localizes index data near infinity via the quotient -theory. The authors prove key results for spaces admitting relative fibred coarse embeddings (RFCE), showing the relative assembly maps are isomorphisms and deriving PSC obstructions that sharpen Gromov–Lawson-type results. They further derive global consequences from relative data, including maximal BC for products of RFCE spaces and demonstrations of subset-sensitivity in the relative conjectures. Overall, the approach yields both conceptual insights and computable tools for PSC obstructions and coarse geometric conjectures in the non-compact setting.

Abstract

In this paper, we employ quotients of Roe algebras as index containers for elliptic differential operators to study the existence problem of Riemannian metrics with positive scalar curvature on non-compact complete Riemannian manifolds. The non-vanishing of such an index locates the precise direction at infinity of the obstructions to positive scalar curvature, and may be viewed as a refinement of the positive scalar curvature problem. To achieve this, we formulate the relative coarse Baum-Connes conjecture and the relative coarse Novikov conjecture, together with their maximal versions, for general metric spaces as a program to compute the -theory of the quotients of the Roe algebras relative to specific ideals. We show that if the metric space admits a relative fibred coarse embedding into Hilbert space or an -space, certain cases of these conjectures can be verified, which yield obstructions to the existence of uniformly positive scalar curvature metrics in specified directions at infinity. As an application, we prove that the maximal coarse Baum-Connes conjecture holds for finite products of certain expander graphs that fail to admit fibred coarse embeddings into Hilbert space.

Paper Structure

This paper contains 17 sections, 20 theorems, 130 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Quotients of Roe algebras and obstruction to PSC metrics
  3. Roe algebras
  4. Coarse groupoid and ideals of Roe algebras
  5. Relative higher index and positive scalar curvature obstruction
  6. Relative higher index theory
  7. Relative coarse assembly maps
  8. Existence of positive scalar curvature at infinity
  9. A remark on general ghostly ideals
  10. Relative fibred coarse embeddings
  11. Relative fibred coarse embedding
  12. Coarsely proper algebras
  13. A conceptual description of uniformly almost flat Bott generators
  14. Applications: from "relative" to "global"
  15. From relative to global
  16. ...and 2 more sections

Key Result

Proposition 2.3

There exists a canonical $*$-isomorphism which extends to $C^*$--somorphisms

Figures (2)

  • Figure 1: Products of two FCE spaces
  • Figure 2: Products of three FCE spaces

Theorems & Definitions (56)

  • Definition 2.1
  • Definition 2.2
  • Conjecture
  • Proposition 2.3: STY2002
  • Definition 2.4
  • Proposition 2.5: CW2004
  • Definition 2.6
  • Remark 2.7
  • Definition 2.8
  • Remark 2.9
  • ...and 46 more