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

Authors

Carmen Rovi (Department of Mathematics, Indiana University, Bloomington, In., U.S.A.)

Shoji Yokura (Department of Mathematics and Computer Science, Graduate School of Science and Engineering, Kagoshima University, Kagoshima, Japan)

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.

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Partially supported by JSPS KAKENHI Grant Number 16H03936.

Received 18 April 2017