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Mathematics, Algebraic Topology, Algebraic Geometry, Geometric Topology
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.
