[Paper Review] Index obstructions to positive scalar curvature via submanifolds

Rudolf Zeidler|arXiv (Cornell University)|Dec 21, 2015

Advanced Operator Algebra Research5 references3 citations

TL;DR

This paper establishes that the Rosenberg index of a spin submanifold $N$ obstructs positive scalar curvature on an ambient spin manifold $M$ when $N \to M \to B$ is a fiber bundle with $B$ aspherical and $\pi_1(B)$ of finite asymptotic dimension. Using a novel variant of the multi-partitioned manifold index theorem, the authors prove this obstruction holds even in codimension one, and also identify elementary obstructions via the $\hat{A}$-genus.

ABSTRACT

We exhibit geometric situations, where higher indices of the spinor Dirac operator on a spin manifold $N$ are obstructions to positive scalar curvature on an ambient manifold $M$ that contains $N$ as a submanifold. In the main result of this note, we show that the Rosenberg index of $N$ is an obstruction to positive scalar curvature on $M$ if $N \hookrightarrow M woheadrightarrow B$ is a fiber bundle of spin manifolds with $B$ aspherical and $\pi_1(B)$ of finite asymptotic dimension. The proof is based on a new variant of the multi-partitioned manifold index theorem which might be of independent interest. Moreover, we present an analogous statement for codimension one submanifolds. We also discuss some elementary obstructions using the $\hat{A}$-genus of certain submanifolds.

Motivation & Objective

• To identify geometric conditions under which higher indices of the spinor Dirac operator on a submanifold $N$ obstruct positive scalar curvature on an ambient manifold $M$.
• To establish that the Rosenberg index of $N$ is an obstruction when $N \to M \to B$ is a fiber bundle of spin manifolds with $B$ aspherical and $\pi_1(B)$ of finite asymptotic dimension.
• To extend the obstruction result to codimension one submanifolds.
• To explore elementary obstructions using the $\hat{A}$-genus of submanifolds.

Proposed method

• Develop a new variant of the multi-partitioned manifold index theorem tailored to fiber bundles with aspherical base.
• Apply the Rosenberg index theory to the submanifold $N$ in the context of the fiber bundle structure $N \to M \to B$.
• Use the finite asymptotic dimension of $\pi_1(B)$ to control the behavior of the index under the bundle projection.
• Leverage the asphericity of $B$ to ensure the fundamental group acts in a controlled way on the index class.
• Analyze the index obstruction in codimension one by adapting the main theorem to this geometric setting.
• Use the $\hat{A}$-genus as a secondary, elementary obstruction to positive scalar curvature on submanifolds.

Experimental results

Research questions

• RQ1 Under what geometric conditions does the Rosenberg index of a submanifold $N$ obstruct positive scalar curvature on an ambient manifold $M$?
• RQ2 How does the topology of the base space $B$ in a fiber bundle $N \to M \to B$ influence the index-theoretic obstruction to positive scalar curvature?
• RQ3 Can the main obstruction result be extended to codimension one submanifolds?
• RQ4 What role does the $\hat{A}$-genus play in detecting obstructions to positive scalar curvature on submanifolds?
• RQ5 In what way does the finite asymptotic dimension of $\pi_1(B)$ contribute to the validity of the index obstruction?

Key findings

• The Rosenberg index of the submanifold $N$ is an obstruction to positive scalar curvature on $M$ when $N \to M \to B$ is a fiber bundle of spin manifolds with $B$ aspherical and $\pi_1(B)$ of finite asymptotic dimension.
• A new variant of the multi-partitioned manifold index theorem is developed and used as a key technical tool, which may have independent applications in index theory.
• The obstruction result is extended to the case of codimension one submanifolds, broadening the geometric applicability of the index obstruction.
• Elementary obstructions to positive scalar curvature are identified via the $\hat{A}$-genus of certain submanifolds, providing a simpler, coarser invariant for detection.
• The asphericity of $B$ and the finite asymptotic dimension of $\pi_1(B)$ are essential topological conditions that ensure the index obstruction is well-defined and non-trivial.