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<p>We establish several non-existence results of positive scalar curvature (PSC) on fiber bundles. We show that under an incompressible condition of the fiber, for <i>X</i><sup><i>m</i></sup>, a Cartan-Hadamard manifold or an aspherical manifold when <i>m</i> = 3, the fiber bundle over <i>X</i><sup><i>m</i></sup>#<i>M</i><sup><i>m</i></sup> with the <i>K</i>(<i>π</i>, 1) fiber and NPSC<sup>+</sup>(a manifold class including enlargeable and Schoen-Yau-Schick ones) fiber, or spin fiber of the non-vanishing Rosenberg index carries no PSC metric, with necessary dimension and spin compatible conditions imposed. Furthermore, we show that under a homotopically nontrivial condition of the fiber, the <i>S</i><sup>1</sup> bundle over a closed 3-manifold admits a PSC metric if and only if its base space does. These partially answer a question of Gromov (2018) and extend some previous results of Hanke et al. (2015) and Zeidler (2017) concerning PSC obstruction on fiber bundles.</p>

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Twisted S1 stability and positive scalar curvature obstruction on fiber bundles

  • 期刊论文

  • 发表日期：

    2025年12月26日

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  • Shihang He

摘要

We establish several non-existence results of positive scalar curvature (PSC) on fiber bundles. We show that under an incompressible condition of the fiber, for Xm, a Cartan-Hadamard manifold or an aspherical manifold when m = 3, the fiber bundle over Xm#Mm with the K(π, 1) fiber and NPSC+(a manifold class including enlargeable and Schoen-Yau-Schick ones) fiber, or spin fiber of the non-vanishing Rosenberg index carries no PSC metric, with necessary dimension and spin compatible conditions imposed. Furthermore, we show that under a homotopically nontrivial condition of the fiber, the S1 bundle over a closed 3-manifold admits a PSC metric if and only if its base space does. These partially answer a question of Gromov (2018) and extend some previous results of Hanke et al. (2015) and Zeidler (2017) concerning PSC obstruction on fiber bundles.