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# Pure and Applied Mathematics Quarterly

## Volume 12 (2016)

### Number 4

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

Pages: 587 – 602

DOI: https://dx.doi.org/10.4310/PAMQ.2016.v12.n4.a7

#### Authors

#### Abstract

I. Hambleton, A. Korzeniewski and A. Ranicki have proved that the signature of a fiber bundle $F \hookrightarrow E \to B$ of closed, connected, compatibly oriented PL manifolds is always multiplicative $\mathrm{mod} \: 4$, i.e. $\sigma (E) \equiv \sigma (F) \sigma (B) \: \mathrm{mod} \: 4$. In this paper, we consider the Hirzebruch $\chi_y$-genera for odd integers $y$ for a smooth fiber bundle $F \hookrightarrow E \to B$ such that $E$, $F$ and $B$ are compact complex algebraic manifolds (in the complex analytic topology, not in the Zariski topology). In particular, if $y = 1$, then $\chi_1$ is the signature $\sigma$. We show that the Hirzebruch $\chi_y$-genera of such a fiber bundle are always multiplicative $\mathrm{mod} \: 4$, i.e. $\chi_y (E) \equiv \chi_y (F) \chi_y (B) \: \mathrm{mod} \: 4$. We also investigate multiplicativity $\mathrm{mod} \: 8$, and show that if $y \equiv 3 \: \mathrm{mod} \: 4$, then $\chi_y (E) \equiv \chi_y (F) \chi_y (B) \: \mathrm{mod} \: 8$ and that in the case when $y \equiv 1 \: \mathrm{mod} \: 4$ the Hirzebruch $\chi_y$-genera of such a fiber bundle is multiplicative $\mathrm{mod} \: 8$ if and only if the signature is multiplicative $\mathrm{mod} \: 8$, and that the non-multiplicativity $\mathrm{modulo} \: 8$ is identified with an Arf–Kervaire invariant.

Partially supported by JSPS KAKENHI Grant Number 16H03936.

Received 18 April 2017

Published 26 July 2018