A comparison of norm maps

TL;DR

This paper develops a spectrum-level norm map in equivariant homotopy theory, compares it with existing constructions, and provides a conceptual understanding of their differences, enhancing the theoretical framework of the field.

Contribution

It introduces a spectrum-level norm map based on algebraic methods and compares it with the Hill-Hopkins-Ravenel construction, clarifying their relationship.

Findings

The spectrum-level norm map aligns with Hill-Hopkins-Ravenel's construction.

Provides a conceptual explanation for differences in norm map definitions.

Enhances understanding of multiplicative norm maps in equivariant homotopy theory.

Abstract

We present a spectrum-level version of the norm map in equivariant homotopy theory based on the algebraic construction in work of Greenleess-May. We show that this new norm map is same as the construction in work Hill-Hopkins-Ravenel on the Kervaire invariant problem. Our comparison of the two norm maps gives a conceptual understanding of the choices inherent in the definition of the multiplicative norm map.