Inspired by Gromov’s work on
Metric inequalities with scalar curvature
we establish band width inequalities for Riemannian bands of the form
, where
is a closed
manifold. We introduce a new class of orientable manifolds we call
filling-enlargeable and
prove: If
is filling-enlargeable and all unit balls in the universal cover of
have volume less
than a constant
,
then
.
We show that if a closed orientable manifold is enlargeable or aspherical, then it is
filling-enlargeable. Furthermore, we establish that whether a closed orientable
manifold is filling-enlargeable or not only depends on the image of the fundamental
class under the classifying map of the universal cover.
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