Volume of Riemannian manifolds, geometric inequalities, and homotopy theory

Alexander I. Suciu; Mikhail G. Katz

Volume of Riemannian manifolds, geometric inequalities, and homotopy theory

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arxiv math/9810172 v2 pith:UX6CRHKV submitted 1998-10-29 math.DG math.AT

Mikhail G. Katz , Alexander I. Suciu This is my paper · ORCID

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keywords everygeometricinequalitiesorientablesystolicvolumeadmitsarbitrarily

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We outline the current state of knowledge regarding geometric inequalities of systolic type, and prove new results, including systolic freedom in dimension 4. Namely, every compact, orientable, smooth 4-manifold X admits metrics of arbitrarily small volume such that every orientable, immersed surface of smaller than unit area is necessarily null-homologous in X.

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Transversal non-Clifford gates on qLDPC codes breaking the $\sqrt{N}$ distance barrier and quantum-inspired geometry with $\mathbb{Z}_2$ systolic freedom
quant-ph 2025-07 conditional novelty 6.0 of 10

A triple homological product of good quantum LDPC codes achieves distance N^(2/3) with transversal CCZ gates and prepares N^(1/3) magic states in a single round.

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