[Paper Review] Positive scalar curvature, higher rho invariants and localization algebras

Zhizhang Xie, Guoliang Yu|arXiv (Cornell University)|Feb 18, 2013

Advanced Operator Algebra Research23 references7 citations

TL;DR

This paper establishes a precise connection between the higher index class of the Dirac operator on a spin manifold with boundary and the higher rho invariant of the boundary Dirac operator, using localization algebras to define and compute the latter. The main result shows that the $K$-theoretic boundary of the higher index class on the manifold equals the higher rho invariant on the boundary, extending work by Piazza and Schick and providing a new framework for studying positive scalar curvature obstructions via noncommutative topology.

ABSTRACT

In this paper, we use localization algebras to study higher rho invariants of closed spin manifolds with positive scalar curvature metrics. The higher rho invariant is a secondary invariant and is closely related to positive scalar curvature problems. The main result of the paper connects the higher index of the Dirac operator on a spin manifold with boundary to the higher rho invariant of the Dirac operator on the boundary, where the boundary is endowed with a positive scalar curvature metric. Our result extends a theorem of Piazza and Schick.

Motivation & Objective

• To extend the higher index theory of Atiyah-Patodi-Singer to manifolds with boundary using localization algebras.
• To define and compute the higher rho invariant of the boundary Dirac operator in the presence of positive scalar curvature.
• To establish a precise relationship between the higher index class on the manifold and the higher rho invariant on the boundary.
• To provide a K-theoretic framework for studying obstructions to positive scalar curvature metrics on manifolds with boundary.

Proposed method

• Uses localization algebras $C^{ullet}_{L,0}(X)^{lat}$ and $C^{ullet}_L(X)^{lat}$ associated with a proper, cocompact isometric action of a discrete group $\Gamma$ on a proper metric space $X$.
• Applies the six-term exact sequence in $K$-theory to the short exact sequence of $C^*$-algebras: $0 \to C^{ullet}_{L,0}(X)^{\Gamma} \to C^{ullet}_L(X)^{\Gamma} \to C^{ullet}(X)^{\Gamma} \to 0$
• Constructs canonical boundary maps $\partial_i$ in $K$-theory to relate $K_*(D^*(X)^\Gamma)$ to $K_{*-1}(C^{ullet}_{L,0}(X)^\Gamma)$
• Defines the higher rho invariant $\rho(D_{\partial M}, h)$ as an element in $K_{m-1}(C^{ullet}_{L,0}(\partial M)^\Gamma)$
• Establishes isomorphisms $\alpha_i: K_i(D^*(X)^\Gamma) \to K_{i-1}(C^\bullet_{L,0}(X)^\Gamma)$ and $\beta_i: K_i(D^*(X)^\Gamma / C^\bullet(X)^\Gamma) \to K_{i-1}(C^\bullet_L(X)^\Gamma)$ via Mayer-Vietoris and five lemma arguments
• Uses the natural identification of $K_*(C^\bullet_L(X)^\Gamma)$ with equivariant $K$-homology $K_*^\Gamma(X)$ to interpret the higher rho invariant in geometric terms.

Experimental results

Research questions

• RQ1 How can the higher rho invariant on the boundary of a spin manifold with positive scalar curvature be related to the higher index class of the Dirac operator on the full manifold?
• RQ2 Can localization algebras provide a robust framework for defining and computing higher rho invariants in the presence of group actions and positive scalar curvature?
• RQ3 Is there a canonical boundary map from the higher index class on a manifold with boundary to the higher rho invariant on the boundary?
• RQ4 Does the higher rho invariant capture the same obstruction to positive scalar curvature as the higher eta invariant, but in a $K$-theoretic setting?
• RQ5 Can the isomorphism between $K_*(D^*(X)^\Gamma)$ and $K_{*-1}(C^\bullet_{L,0}(X)^\Gamma)$ be used to define a canonical higher rho invariant that matches existing definitions?

Key findings

• The higher index class $\mathrm{Ind}(D_M)$ of the Dirac operator on a spin manifold $M$ with boundary $\partial M$ maps via a canonical boundary map to the higher rho invariant $\rho(D_{\partial M}, h)$ in $K_{m-1}(C^\bullet_{L,0}(\partial M)^\Gamma)$
• The map $\partial_1: K_1(C^\bullet_L(M)^\Gamma) \to K_0(C^\bullet_{L,0}(M)^\Gamma)$ identifies the boundary of the higher index class with the higher rho invariant on the boundary
• The isomorphism $\alpha_1: K_1(D^*(\partial M)^\Gamma) \to K_0(C^\bullet_{L,0}(\partial M)^\Gamma)$ provides a $K$-theoretic realization of the higher rho invariant
• The higher rho invariant $\rho(D_{\partial M}, h)$ is equivalent to the definition of Higson and Roe [18, Definition 7.1] via the isomorphism $\alpha_i$
• The construction is natural and commutes with the six-term exact sequence in $K$-theory, ensuring consistency across the framework
• The result generalizes Piazza and Schick's Theorem 1.17 by extending the boundary relation to the setting of group actions and localization algebras.