Explicit formulas for the Hattori-Stong theorem and applications

TL;DR

This paper derives explicit formulas for Chern class coefficients in the Hattori-Stong theorem, providing new conditions on signatures of stably almost-complex manifolds and applications to rational projective planes.

Contribution

It presents an explicit combinatorial formula for Hattori-Stong coefficients and introduces an evenness condition for signatures based on Chern numbers.

Findings

Signature of certain stably almost-complex manifolds is even under specific Chern number conditions

Rules out existence of such structures on rational projective planes

Provides explicit formulas for coefficients in Hattori-Stong integrability conditions

Abstract

We employ combinatorial techniques to present an explicit formula for the coefficients in front of Chern classes involving 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 showed that the signature of a $2n$-dimensional stably almost-complex manifold whose possibly nonzero Chern numbers being $c_n$ and $c_ic_{n-i}$ is even, which particularly rules out the existence of such structure on rational projective planes. Some other related results and remarks are also discussed in this article.