Theoretical study demonstrates sharp band width estimates in fiber bundle families, highlighting the validity of Gromov's conjecture under infinite family A-hat area.
We extend Gromov’s conjecture on the sharp width estimate for Riemannian bands with positive scalar curvature to the family case and prove that it holds for fiber bundles with infinite family  -area. The method we employ is based on Dirac operators and the family index theory. Our proof relies on a product formula for index bundles established in this paper.
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