Hirzebruch χy\chi_y-genera modulo 88 of fiber bundles for odd integers yy

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

Grothendieck groups and a categorification of additive invariants

A topologically-invariant and additive homology class is mostly not a natural transformation as it is. In this paper we discuss turning such a homology class into a natural transformation; i.e., a "categorification" of it. In a general categorical set-up we introduce a generalized relative Grothendieck group from a cospan of functors of categories and also consider a categorification of additive invariants on objects. As an example, we obtain a general theory of characteristic homology classes of singular varieties.Comment: 27 pages, to appear in International J. Mathematic

Inclusion-exclusion and Segre classes

We propose a variation of the notion of Segre class, by forcing a naive `inclusion-exclusion' principle to hold. The resulting class is computationally tractable, and is closely related to Chern-Schwartz-MacPherson classes. We deduce several general properties of the new class from this relation, and obtain an expression for the Milnor class of a scheme in terms of this class.Comment: 8 page

Hirzebruch-Milnor classes and Steenbrink spectra of certain projective hypersurfaces

We show that the Hirzebruch-Milnor class of a projective hypersurface, which gives the difference between the Hirzebruch class and the virtual one, can be calculated by using the Steenbrink spectra of local defining functions of the hypersurface if certain good conditions are satisfied, e.g. in the case of projective hyperplane arrangements, where we can give a more explicit formula. This is a natural continuation of our previous paper on the Hirzebruch-Milnor classes of complete intersections.Comment: 15 pages, Introduction is modifie