Hirzebruch $\chi_y$-genera modulo $8$ of fiber bundles for odd integers $y$

TL;DR

This paper studies the multiplicativity properties of Hirzebruch $oldsymbol{ ext{chi}_y}$-genera modulo 8 for fiber bundles of complex algebraic manifolds, revealing conditions related to the signature and Arf-Kervaire invariant.

Contribution

It extends known results on multiplicativity modulo 4 to modulo 8 for specific values of y, linking these properties to the signature and Arf-Kervaire invariant.

Findings

Hirzebruch $ ext{chi}_y$-genera are multiplicative modulo 4.

For y ≡ 3 mod 4, they are multiplicative modulo 8.

For y ≡ 1 mod 4, multiplicativity modulo 8 depends on the signature and relates to the Arf-Kervaire invariant.

Abstract

I. Hambleton, A. Korzeniewski and A. Ranicki proved that the signature of a fibre bundle of closed, connected, compatibly oriented PL manifolds is always multiplicative modulo 4. In this paper, we consider the Hirzebruch $\chi_y$-genera for odd integers $y$ for a smooth fiber bundle such that the base, fibre, and total space are compact complex algebraic manifolds (in the complex analytic topology, not in the Zariski topology). We show that the Hirzebruch $\chi_y$-genera of such a fibre bundle are always multiplicative modulo 4. We also investigate multiplicativity modulo 8 and show that if $y$ is congruent to 3 modulo 4, then the $\chi_y$-genera are multiplicative modulo 8. We also show that when $y$ is congruent to 1 modulo 4, the Hirzebruch $\chi_y$-genera of such a fiber bundle are multiplicative modulo 8 if and only if the signature is multiplicative modulo 8, and that the non-multiplicativity modulo 8, in this case, is identified with an Arf-Kervaire invariant.