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arXiv · 2508.14791

Small scale index theory, scalar curvature, and Gromov's simplicial norms

Qiaochu Ma·Guoliang Yu
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Abstract

In this article, we study the topological complexity of manifolds with a lower scalar curvature bound. We introduce a small scale index theorem to establish an upper bound for Gromov's simplicial norm of the Poincar\'e dual of the A-hat class for manifolds with spin universal covering, in terms of a scalar curvature lower bound, volume upper bound, and injectivity radius lower bound of the universal covering. This result can be viewed both as a generalization of Lichnerowicz vanishing theorem and as a scalar curvature analogue to Cheeger finiteness theorem.

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Qiaochu Ma, Guoliang Yu. 2025-08-20. Small scale index theory, scalar curvature, and Gromov's simplicial norms. https://arxiv.org/abs/2508.14791

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