Explicit formulas for the Hattori–Stong theorem and applications

Li, Ping; Lin, Wangyang

doi:10.2140/agt.2026.26.735

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Explicit formulas for the Hattori–Stong theorem and applications

Ping Li and Wangyang Lin

Algebraic & Geometric Topology 26 (2026) 735–750

DOI: 10.2140/agt.2026.26.735

Abstract

We employ combinatorial techniques to present an explicit formula for the coefficients of Chern classes involved in the Hattori–Stong integrability conditions. We also give an evenness condition for the signature of stably almost-complex manifolds in terms of Chern numbers. As an application, it can be shown that the signature of a $2 n$ -dimensional stably almost-complex manifold whose possibly nonzero Chern numbers being $c_{n}$ and $c_{i} c_{n - i}$ is even, which particularly rules out the existence of such structure on rational projective planes. Some other related results and remarks are also discussed.

Keywords

stably almost-complex structure, Chern number, Hattori–Stong theorem, Hirzebruch signature formula, the Stirling number of the second kind, Bernoulli number, rational projective plane

Mathematical Subject Classification

Primary: 05E05, 19L64, 32Q60, 57R20

References

Publication

Received: 14 October 2024

Revised: 1 March 2025

Accepted: 15 March 2025

Published: 11 February 2026

Authors

Ping Li
School of Mathematical Sciences Fudan University Shanghai China

Wangyang Lin
School of Mathematical Sciences Fudan University Shanghai China

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Volume 26, issue 2 (2026)