[Paper Review] Infinite connected sums, K-area and positive scalar curvature

Levi Lopes de Lima|arXiv (Cornell University)|Aug 17, 2004

Advanced Operator Algebra Research10 references3 citations

TL;DR

This paper establishes a new obstruction to positive scalar curvature on infinite connected sums of spin manifolds using bounded $K$-area invariants. By showing that the $K$-area of such a sum becomes infinite when attaching a manifold with infinite $K$-area, it proves that no metric of positive scalar curvature can exist in the bounded geometry class, generalizing earlier results based on index theory.

ABSTRACT

Whyte used the index theory of Dirac operators and Block-Weiberger uniformly finite homology to show that certain infinite connected sums do not carry a metric with nonnegative scalar curvature in their bounded geometry class. His proof uses a coarse version of the $\hat{A}$-class to obstruct such metrics. In this note we prove a version of Whyte's result where a variant of the notion of infinite $K$-area, originally due to Gromov, is used to obstruct metrics with positive scalar curvature.

Motivation & Objective

• To extend existing obstructions to positive scalar curvature on infinite connected sums beyond index-theoretic methods.
• To establish that bounded $K$-area invariants provide a stronger, more general obstruction in the bounded geometry setting.
• To prove that infinite $K$-area in the sum implies non-existence of positive scalar curvature metrics.
• To show that $K_{ m area}^b(X) = +∞$ implies no metric with scalar curvature non-positive outside arbitrarily large compact sets.
• To demonstrate that the $K$-area obstruction applies in all even dimensions, not just multiples of 4, unlike the $˂ A$-class obstruction.

Proposed method

• Use of Gromov's notion of $K$-area, adapted to bounded geometry via the invariant $K_{ m area}^b(X)$, which is stable under bounded diffeomorphisms.
• Decomposition of the bounded geometry category ${\mathcal{B}\mathcal{G}}_{n}$ into amenable ($H_0^{\rm uf}(X) \neq 0$) and non-amenable components.
• Proof that $K_{ m area}^b(Y \sharp_S M) = +\infty$ when $K_{ m area}(M) = +\infty$ and $Y \in \mathcal{B}\mathcal{G}_{2k}^A$ with finite $K_{ m area}^b(Y)$.
• Application of Adams operations on vector bundles to analyze Chern character classes and derive vanishing results for bounded Chern numbers.
• Use of spectral theory and curvature bounds to estimate index and $L^2$-norms of Dirac operators on large domains.
• Reduction of the problem to the vanishing of bounded Chern characters and classes via universal polynomial relations.

Experimental results

Research questions

• RQ1 Can $K$-area invariants be used to obstruct positive scalar curvature on infinite connected sums in bounded geometry?
• RQ2 Does the $K$-area of an infinite connected sum remain finite when attaching a manifold with infinite $K$-area?
• RQ3 Is there a geometric obstruction to positive scalar curvature that is stronger than the $˂ A$-class obstruction and applies in all even dimensions?
• RQ4 Can the $K$-area invariant detect the non-existence of positive scalar curvature metrics even when the $˂ A$-class vanishes?
• RQ5 What is the role of uniformly finite homology and bounded geometry in the stability of $K$-area under infinite connected sums?

Key findings

• If $Y \in \mathcal{B}\mathcal{G}_{2k}^A$ has finite $K_{\rm area}^b(Y)$ and $[S] \neq 0$ in $H_0^{\rm uf}(Y)$, then $Y \sharp_S M$ has $K_{\rm area}^b(Y \sharp_S M) = +\infty$ whenever $K_{\rm area}(M) = +\infty$.
• A manifold $X \in \mathcal{B}\mathcal{G}_{2k}$ with $K_{\rm area}^b(X) = +\infty$ cannot carry a metric whose scalar curvature is non-positive outside arbitrarily large compact sets.
• In particular, such a manifold admits no metric of positive scalar curvature in its bounded geometry class.
• The $K$-area obstruction applies in all even dimensions $2k$, not just multiples of 4, making it more general than the $˂ A$-class obstruction.
• The vanishing of all bounded Chern numbers ${c}_I^b({\mathcal{E}})$ follows from the $K$-area finiteness, implying $\mathcal{E}$ is $b$-homologically trivial.
• The proof relies on Adams operations and curvature-controlled spectral estimates to show that the reduced Chern character classes must vanish, leading to contradiction if $K_{\rm area}^b(X) < \infty$.