CRone.abstract
The Dyer-Lashof Algebra in Bordism
Terrence Bisson and Andr\'e Joyal
(June 1995. To Appear, C.R.Math.Rep.Acad.Sci.Canada)
We present a theory of Dyer-Lashof operations in unoriented bordism
(the canonical splitting $N_*(X)\simeq N_*\otimes H_*(X)$, where
$N_*(\ )$ is unoriented bordism and $H_*(\ )$ is homology mod 2,
does not respect these operations). For any finite covering space we
define a ``polynomial functor'' from the category of topological
spaces to itself. If the covering space is a closed manifold
we obtain an operation defined on the bordism of any $E_\infty$-space.
A certain sequence of operations called squaring operations are
defined from two-fold coverings; they satisfy the Cartan formula and
also a generalization of the Adem relations that is formulated by
using Lubin's theory of isogenies of formal group laws. We call a
ring equipped with such a sequence of squaring operations a
$D$-{\it ring}, and observe that the bordism ring of any free
$E_\infty$-space is free as a $D$-ring. In particular, the bordism
ring of finite covering manifolds is the free $D$-ring on one generator.
In a second compte-rendu we discuss the (Nishida) relations between
the Landweber-Novikov and the Dyer-Lashof operations, and show how
to represent the Dyer-Lashof operations in terms of their actions on
the characteristic numbers of manifolds.