Twisted $S^1$ stability and positive scalar curvature obstruction on fiber bundles

Abstract

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 $X^m$ a Cartan-Hadamard manifold or an aspherical manifold when $m=3$, the fiber bundle over $X^m\#M^m$ with the $K(\pi,1)$ fiber, $NPSC^+$(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 $S^1$ 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 and extend some previous results of Hanke, Schick and Zeidler concerning PSC obstruction on fiber bundles.