[Paper Review] A relative higher index theorem, diffeomorphisms and positive scalar curvature

Zhizhang Xie, Guoliang Yu|arXiv (Cornell University)|Apr 16, 2012

Advanced Operator Algebra Research16 references3 citations

TL;DR

This paper establishes a relative higher index theorem for complete spin manifolds with positive scalar curvature at infinity, introducing a secondary higher index class that obstructs two positive scalar curvature metrics on a closed manifold and its Galois cover from being connected in the space of such metrics. When one metric is induced by a diffeomorphism of the manifold, the paper provides a computable formula for this index class, proving it lies in the image of the Baum-Connes assembly map.

ABSTRACT

We prove a general relative higher index theorem for complete manifolds with positive scalar curvature towards infinity. We apply this theorem to study Riemannian metrics of positive scalar curvature on manifolds. For every two metrics of positive scalar curvature on a closed manifold and a Galois cover of the manifold, we define a secondary higher index class. Non-vanishing of this higher index class is an obstruction for the two metrics to be in the same connected component of the space of metrics of positive scalar curvature. In the special case where one metric is induced from the other by a diffeomorphism of the manifold, we obtain a formula for computing this higher index class. In particular, it follows that the higher index class lies in the image of the Baum-Connes assembly map.

Motivation & Objective

• To develop a relative higher index theorem for complete manifolds with positive scalar curvature at infinity.
• To define a secondary higher index class that obstructs two positive scalar curvature metrics on a closed manifold and its Galois cover from being in the same connected component of the space of such metrics.
• To derive a computable formula for this index class in the case where one metric is induced by a diffeomorphism of the manifold.
• To show that the resulting higher index class lies in the image of the Baum-Connes assembly map, linking it to deep conjectures in noncommutative geometry.

Proposed method

• Construct a relative higher index theorem using Dirac operators on Galois covers of complete spin manifolds with positive scalar curvature outside compact sets.
• Define the relative index class via gluing two manifolds along a hypersurface, producing a new manifold whose Dirac operator's higher index class equals the difference of the original index classes.
• Use the lifting of isometries and bundle isometries between spinor bundles on the ends of the manifolds to ensure compatibility in the gluing process.
• Apply the theory of elliptic pseudodifferential operators and Sobolev norms to control the regularity and boundedness of sections in the index computation.
• Leverage the Baum-Connes assembly map to show that the secondary index class lies in its image, particularly in the diffeomorphism-induced case.
• Generalize results from the complex to the real $C^*$-algebraic setting, though proofs are presented for the complex case with remarks on real case adaptation.

Experimental results

Research questions

• RQ1 What is the obstruction to two positive scalar curvature metrics on a closed manifold and its Galois cover being connected in the space of such metrics?
• RQ2 How can a relative higher index theorem be formulated for complete manifolds with positive scalar curvature at infinity?
• RQ3 Can the secondary higher index class be explicitly computed when one metric is induced by a diffeomorphism of the manifold?
• RQ4 Does the secondary higher index class lie in the image of the Baum-Connes assembly map in the diffeomorphism-induced case?
• RQ5 How do the geometric and analytic properties of the Dirac operator and its higher index behave under gluing constructions involving hypersurfaces and Galois covers?

Key findings

• The relative higher index theorem establishes that the higher index class of the glued manifold $X_2$ equals the difference of the higher index classes of the original manifolds: $\textup{Ind}(D_2) = \textup{Ind}(D_0) - \textup{Ind}(D_1)$.
• The secondary higher index class associated with two positive scalar curvature metrics on a closed manifold and its Galois cover is a topological obstruction to their connectedness in the space of such metrics.
• When one metric is induced from the other by a diffeomorphism, the secondary index class is computable and lies in the image of the Baum-Connes assembly map.
• The theorem applies to both real and complex $C^*$-algebras, with the complex case treated in detail and real case modifications indicated.
• The proof relies on the theory of elliptic pseudodifferential operators, Sobolev norms, and restriction maps on $\mathcal{A}$-bundles, with key estimates established via parametrix constructions.
• The restriction map on Sobolev spaces satisfies the inequality $\langle\langle R\sigma,R\sigma\rangle\rangle_{s-k/2} \leq C\langle\langle\sigma,\sigma\rangle\rangle_s$ for $s > k/2$, which is essential for regularity control in the index computation.