Hirzebruch Xy-genera modulo 8 of fiber bundles for odd integers y

Rovi, Carmen; Yokura, Shoji

doi:10.4310/PAMQ.2016.v12.n4.a7

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Hirzebruch X_y-genera modulo 8 of fiber bundles for odd integers y

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Rovi, Carmen
Max Planck Institute for Mathematics, Max Planck Society;

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http://dx.doi.org/10.4310/PAMQ.2016.v12.n4.a7
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https://doi.org/10.48550/arXiv.1609.02118
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Citation

Rovi, C., & Yokura, S. (2016). Hirzebruch X_y-genera modulo 8 of fiber bundles for odd integers y. Pure and Applied Mathematics Quarterly, 12, 587-602. doi:10.4310/PAMQ.2016.v12.n4.a7.

Cite as: https://hdl.handle.net/21.11116/0000-0003-D001-0

Abstract

I. Hambleton, A. Korzeniewski and A. Ranicki have proved that the signature of a fiber bundle F↪E→B of closed, connected, compatibly oriented PL manifolds is always multiplicative mod4, i.e. σ(E)≡σ(F)σ(B)mod4. In this paper, we consider the Hirzebruch χy-genera for odd integers y for a smooth fiber bundle F↪E→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 χ1 is the signature σ. We show that the Hirzebruch χy-genera of such a fiber bundle are always multiplicative mod4, i.e. χy(E)≡χy(F)χy(B)mod4. We also investigate multiplicativity mod8, and show that if y≡3mod4, then χy(E)≡χy(F)χy(B)mod8 and that in the case when y≡1mod4 the Hirzebruch χy-genera of such a fiber bundle is multiplicative mod8 if and only if the signature is multiplicative mod8, and that the non-multiplicativity modulo8 is identified with an Arf–Kervaire invariant.