分野別セミナー

Generating series are used in many area of mathematics, including algebraic topology. In this talk we show how in
some cases, generating series can be used not only to organize and simplify intricate formulae, but also to give
topological insights and more conceptual proofs.

Among others, we will treat the examples of Bisson-Joyal identity for the Adem relations, Ravenel-Wilson main
relation in the Hopf ring for complex oriented cohomology theory, as well as relations among universal characteristic
classes for surface bundles introduced by Randal-WIlliams.

The (mod 2) Steenrod algebra is the algebra of cohomology operations. It is generated by operations $Sq^i$, subject to relations involving binomial coefficients, called Adem relations. Bisson-Joyal express it as a very simple equality of
the form $Q(s)Q(t)=Q(t)Q(s)$.

The Ravenel-Wilson main relation is a relation among elements in the Hopf ring of the $E$-homology of the infinite
loop spaces representing $F$-cohomology, where both $E$ and $F$ are multiplicative complex oriented spectra, that
involves the formal group laws of the both theories. In general case, it would be almost impossible to describe the
identity without the help of generating series. A degenerate case of this relation, when $E=F=HZ$, gives the
Hopf algebra structure of $H_*(CP^{\infty})$.

The universal characteristic classes of the surface bundle is defined using certain Madsen-TIllmann spectrum $MTO(2)$, closely related to the mapping class group of unoriented surfaces. The splitting of $MTO(2)$ (joint work with Hadi Zare) allows us to determine a complete set of relations among these classes. A different point of view
allows us to organise these relations using generating series.
Matters related to this splitting will be discussed in the talk given at Fukuoka University on Monday.