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Functoriality for higher rho invariants of elliptic operators

Hao Guo, Zhizhang Xie, Guoliang Yu

Abstract

Let $N$ be a closed spin manifold with positive scalar curvature and $D_N$ the Dirac operator on $N$. Let $M_1$ and $M_2$ be two Galois covers of $N$ such that $M_2$ is a quotient of $M_1$. Then the quotient map from $M_1$ to $M_2$ naturally induces maps between the geometric $C^*$-algebras associated to the two manifolds. We prove, by a finite-propagation argument, that the \emph{maximal} higher rho invariants of the lifts of $D_N$ to $M_1$ and $M_2$ behave functorially with respect to the above quotient map. This can be applied to the computation of higher rho invariants, along with other related invariants.

Functoriality for higher rho invariants of elliptic operators

Abstract

Let be a closed spin manifold with positive scalar curvature and the Dirac operator on . Let and be two Galois covers of such that is a quotient of . Then the quotient map from to naturally induces maps between the geometric -algebras associated to the two manifolds. We prove, by a finite-propagation argument, that the \emph{maximal} higher rho invariants of the lifts of to and behave functorially with respect to the above quotient map. This can be applied to the computation of higher rho invariants, along with other related invariants.

Paper Structure

This paper contains 21 sections, 22 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Geometric $C^*$-algebras
  5. Geometric setup
  6. Folding maps
  7. Definition of the folding map
  8. Description of the folding map on invariant sections
  9. Induced maps on geometric $C^*$-algebras and $K$-theory
  10. Functional calculus and the wave operator
  11. Functional calculus on the maximal Roe algebra
  12. The wave operator
  13. Functoriality for the higher index
  14. Higher index
  15. Functoriality
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $N$ be a closed, spin Riemannian manifold with positive scalar curvature. Let $D_N$ be the Dirac operator on $N$. Let $M_1$ and $M_2$ be Galois covers of $M$ with deck transformation groups $\Gamma_1$ and $\Gamma_2\cong\Gamma_1/H$ respectively, for a normal subgroup $H$ of $\Gamma_1$. Let $D_1$ where $(\Psi_{L,0})_*$ is the map on $K$-theory induced by $\Psi_{L,0}$ and $\rho_{\textnormal{max}

Theorems & Definitions (55)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3: Yu
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • Remark 2.7
  • Definition 2.8
  • Remark 2.9
  • ...and 45 more