Explicit formulas for the Hattori-Stong theorem and applications

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.

💡 Research Summary

The paper “Explicit formulas for the Hattori‑Stong theorem and applications” develops concrete, closed‑form expressions for the coefficients that appear in the Hattori‑Stong integrality conditions for Chern classes, and uses these formulas to obtain new parity results for the signature of stably almost‑complex manifolds.

Background and motivation.
For a complex vector bundle (E) over a space (X) the operations (\gamma_k) in K‑theory are defined by (\gamma_k(eE)=) the coefficient of (t^k) in (\frac{\wedge_t(1-eE)}{\wedge_t(1)}). The Atiyah–Singer index theorem implies that the numbers
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