Nishida Relations in Bordism and Homology (extended abstract)

TL;DR

This paper explores Nishida relations connecting Dyer-Lashof and Landweber-Novikov operations within unoriented bordism, providing algebraic representations of covering manifolds via homology characteristic numbers.

Contribution

It introduces the relations between Dyer-Lashof and Landweber-Novikov operations and applies them to represent the algebra of covering manifolds.

Findings

Established Nishida relations between key operations

Represented algebra of covering manifolds using homology numbers

Extended previous work on Dyer-Lashof operations in bordism

Abstract

This is the second of a series of Compte Rendus. In the first [1] we have presented a theory of Dyer-Lashof operations in unoriented bordism. Here we shall discuss the (Nishida) relations between Dyer-Lashof and Landweber-Novikov operations. They are used to represent the algebra $N_*\Sigma$ of covering manifolds in terms of their homology characteristic numbers. The proofs are based on the properties of the covering space operations and the notions of D-ring and Q-ring introduced in [1].