Band Width and the Rosenberg Index

Abstract

Abstract A Riemannian manifold is said to have infinite $\mathcal {K}\mathcal {O}$-width if it admits an isometric immersion of an arbitrarily wide Riemannian band whose inward boundary has non-trivial higher index. In this paper, we prove that if a closed spin manifold has infinite $\mathcal {K}\mathcal {O}$-width, then its Rosenberg index does not vanish. This gives a positive answer to a conjecture by Zeidler. We also prove its “multi-dimensional” generalization; if a closed spin manifold admits an isometric immersion of an arbitrarily wide cube-like domain whose lowest dimensional corner has non-trivial higher index, then the Rosenberg index of $M$ does not vanish.