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.
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