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Index of Equivariant Callias-Type Operators and Invariant Metrics of Positive Scalar Curvature

Hao Guo

Abstract

We formulate, for any Lie group G acting isometrically on a manifold M, the general notion of a G-equivariant elliptic operator that is invertible outside of a G-cocompact subset of M. We prove a version of the Rellich lemma for this setting and use this to define the equivariant index of such operators. We show that G-equivariant Callias-type operators are self-adjoint, regular, and hence equivariantly invertible at infinity. Such operators explicitly arise from a pairing of the Dirac operator with the equivariant Higson corona. We apply the theory developed herein to obtain an obstruction to positive scalar curvature metrics on non-cocompact manifolds.

Index of Equivariant Callias-Type Operators and Invariant Metrics of Positive Scalar Curvature

Abstract

We formulate, for any Lie group G acting isometrically on a manifold M, the general notion of a G-equivariant elliptic operator that is invertible outside of a G-cocompact subset of M. We prove a version of the Rellich lemma for this setting and use this to define the equivariant index of such operators. We show that G-equivariant Callias-type operators are self-adjoint, regular, and hence equivariantly invertible at infinity. Such operators explicitly arise from a pairing of the Dirac operator with the equivariant Higson corona. We apply the theory developed herein to obtain an obstruction to positive scalar curvature metrics on non-cocompact manifolds.

Paper Structure

This paper contains 18 sections, 47 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. Notation and Terminology
  3. Sobolev Modules
  4. Boundedness and Adjointability
  5. An Equivariant Rellich Lemma
  6. $G$-invertibility at Infinity
  7. Equivariant Fredholmness
  8. A Simplified Definition of $F$
  9. $G$-invertible Operators as $KK$-elements
  10. $G$-Callias-type Operators
  11. Positivity of $B^2+f$
  12. Essential Self-adjointness and Regularity of $B$
  13. $G$-invertibility at Infinity of $B$
  14. The Endomorphism $\Phi$ and the Higson Corona
  15. Constructing $\Phi$
  16. ...and 3 more sections

Key Result

Lemma 3.2

For each $i\geq 0$, the $C_c^\infty(G)$-valued inner product $\langle\,\,,\,\,\rangle_i$ on the pre-Hilbert $C_c^\infty(G)$-module $C_c^{\infty,i}(E)$ is positive in $C^*(G)$.

Theorems & Definitions (96)

  • Definition 3.1
  • Lemma 3.2
  • proof
  • Definition 3.3
  • Remark 3.4
  • Proposition 3.5
  • Remark 3.6
  • Lemma 3.7
  • proof
  • proof : Proof of Proposition \ref{['prop:boundedhilbert']}
  • ...and 86 more