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 w...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(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 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(2018)and extend some previous results of Hanke et al.(2015)and Zeidler(2017)concerning PSC obstruction on fiber bundles.展开更多
基金supported by National Key R&D Program of China(Grant No.2020YFA0712800)。
文摘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(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 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(2018)and extend some previous results of Hanke et al.(2015)and Zeidler(2017)concerning PSC obstruction on fiber bundles.
