Pure and Applied Mathematics Quarterly

Volume 19 (2023)

Number 6

Special Issue in honor of Professor Blaine Lawson’s 80th birthday

Guest Editors: Shiu-Yuen Cheng, Paulo Lima-Filho, and Stephen Shing-Toung Yau

Covering complexity, scalar curvature, and quantitative $K$-theory

Pages: 2951 – 2972

DOI: https://dx.doi.org/10.4310/PAMQ.2023.v19.n6.a13

Authors

Hao Guo (Yau Mathematical Sciences Center, Tsinghua University, Beijing, China)

Guoliang Yu (Department of Mathematics, Texas A&M University, College Station, Tx., U.S.A.)

Abstract

We establish a relationship between a certain notion of covering complexity of a Riemannian spin manifold and positive lower bounds on its scalar curvature. This makes use of a pairing between quantitative operator $K$-theory and Lipschitz topological $K$-theory, combined with an earlier vanishing theorem for the quantitative higher index.

Full Text (PDF format)

The second author is partially supported by NSF grants 1700021 and 2000082, and by the Simons Fellows Program.

Received 28 March 2022

Received revised 25 December 2022

Accepted 12 February 2023

Published 30 January 2024