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Positive scalar curvature and exotic structures on simply connected four manifolds

Aditya Kumar, Balarka Sen

TL;DR

We address Gromov's band width inequality and Rosenberg's $S^1$-stability in dimension four for simply connected smooth 4-manifolds, noting classical 4D counterexamples from Seiberg–Witten theory but proving the width bound and $S^1$-stability when viewed up to homeomorphism and extending related results to certain non-simply connected cases. The core method combines a Schoen–Yau–type descent via separating $\mu$-bubbles in long PSC bands with a 5D cobordism/surgery framework (including Wall's normal maps) to transfer PSC from separating hypersurfaces to stabilized 4-manifolds, and then invokes the Gromov–Lawson–Schoen–Yau surgery results together with the Gromov–Lawson–Stolz 4D classification up to homeomorphism. In particular, the paper proves a width bound $\mathrm{width}(X,g)\le \frac{2\pi}{5\kappa}$ for PSC obstructions and establishes that $M$ is PSC up to homeomorphism if and only if $M\times S^1$ is PSC, with the dimension-4 GL-L-S classification extended in this setting. These results yield concrete corollaries for stabilizations by $S^2\times S^2$ and highlight that the observed 4D pathologies are intricately linked to smooth structure, while the PSC-band theory persists at the level of homeomorphism classes.

Abstract

We address Gromov's band width inequality and Rosenberg's $S^1$-stability conjecture for simply connected smooth four manifolds. Both results are known to be false in dimension 4 due to counterexamples based on Seiberg-Witten invariants. Nevertheless we show that both of these results hold upon considering simply connected smooth four manifolds up to homeomorphism. We also obtain a related result for non-simply connected smooth four manifolds.

Positive scalar curvature and exotic structures on simply connected four manifolds

TL;DR

We address Gromov's band width inequality and Rosenberg's -stability in dimension four for simply connected smooth 4-manifolds, noting classical 4D counterexamples from Seiberg–Witten theory but proving the width bound and -stability when viewed up to homeomorphism and extending related results to certain non-simply connected cases. The core method combines a Schoen–Yau–type descent via separating -bubbles in long PSC bands with a 5D cobordism/surgery framework (including Wall's normal maps) to transfer PSC from separating hypersurfaces to stabilized 4-manifolds, and then invokes the Gromov–Lawson–Schoen–Yau surgery results together with the Gromov–Lawson–Stolz 4D classification up to homeomorphism. In particular, the paper proves a width bound for PSC obstructions and establishes that is PSC up to homeomorphism if and only if is PSC, with the dimension-4 GL-L-S classification extended in this setting. These results yield concrete corollaries for stabilizations by and highlight that the observed 4D pathologies are intricately linked to smooth structure, while the PSC-band theory persists at the level of homeomorphism classes.

Abstract

We address Gromov's band width inequality and Rosenberg's -stability conjecture for simply connected smooth four manifolds. Both results are known to be false in dimension 4 due to counterexamples based on Seiberg-Witten invariants. Nevertheless we show that both of these results hold upon considering simply connected smooth four manifolds up to homeomorphism. We also obtain a related result for non-simply connected smooth four manifolds.
Paper Structure (17 sections, 19 theorems, 38 equations)

This paper contains 17 sections, 19 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Counterexamples to $S^1$-stability in dimension $4$
  3. Statements of main results
  4. Idea of proof
  5. Outline of paper
  6. Separating $\mu$-bubbles and width inequality
  7. Background
  8. Preliminary definitions and properties
  9. Descent argument
  10. Surgery arguments in dimension $4$
  11. Preliminaries on cobordism and handle decompositions
  12. Trading handles in cobordisms of dimension $5$
  13. Surgery for normal maps in dimension $4$
  14. Obtaining a PSC metric on a stabilization
  15. Width inequality and $S^1$-stability in dimension $4$
  16. ...and 2 more sections

Key Result

Theorem 3

For $n\geq 5$, let $M^n$ be a closed simply connected manifold. If $M$ is not spin, then it admits a metric of positive scalar curvature glsc. If $M$ is spin, then it admits a metric of positive scalar curvature if and only if $\hat{\alpha}(M)=0$stolz.

Theorems & Definitions (49)

  • Conjecture 1: $S^1$-stability
  • Conjecture 2: Width inequality
  • Theorem 3: Gromov-Lawson, Stolz glscstolz
  • Remark 4
  • Example : Counterexample to $S^1$ stability in $n=4$
  • Example : Direct argument that $V_5 \times S^1$ is PSC
  • Remark 5
  • Definition 6
  • Theorem A: Width inequality in dimension $4$
  • Theorem B: $S^1$-stability in dimension $4$
  • ...and 39 more