We exhibit geometric situations where higher indices of the spinor Dirac operator on a spin manifold are obstructions to positive scalar curvature on an ambient manifold that contains as a submanifold. In the main result of this note, we show that the Rosenberg index of is an obstruction to positive scalar curvature on if is a fiber bundle of spin manifolds with aspherical and of finite asymptotic dimension. The proof is based on a new variant of the multipartitioned manifold index theorem which might be of independent interest. Moreover, we present an analogous statement for codimension-one submanifolds. We also discuss some elementary obstructions using the -genus of certain submanifolds.
"An index obstruction to positive scalar curvature on fiber bundles over aspherical manifolds." Algebr. Geom. Topol. 17 (5) 3081 - 3094, 2017. https://doi.org/10.2140/agt.2017.17.3081