The Dyer-Lashof algebra in bordism (extended abstract)

TL;DR

This paper develops a theory of Dyer-Lashof operations in unoriented bordism, introducing new algebraic structures called D-rings, and explores their relations with other operations and characteristic numbers in topology.

Contribution

It introduces the concept of D-rings with squaring operations in bordism, extending the algebraic framework of Dyer-Lashof operations to unoriented bordism.

Findings

Bordism rings of free E-infinity spaces are free D-rings.

Defined squaring operations satisfying Cartan and Adem relations.

Connected Dyer-Lashof operations with characteristic numbers of manifolds.

Abstract

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-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.