Multiplicative 2-cocycles at the prime 2

TL;DR

This paper computes the 2-primary component of the scheme of symmetric multiplicative 2-cocycles using Lubin-Tate cohomology, revealing structures relevant to homotopy theory and connective K-theory.

Contribution

It provides a complete computation of the scheme of symmetric multiplicative 2-cocycles at prime 2, building on previous classifications and extending applications in homotopy theory.

Findings

Computed the 2-primary component of the scheme of symmetric multiplicative 2-cocycles.

Connected the results to structures in homotopy theory and connective K-theory.

Extended previous low-order computations to a full classification.

Abstract

Using a previous classification result on symmetric additive 2-cocycles, we collect a variety of facts about the Lubin-Tate cohomology of formal groups to compute the 2-primary component of the scheme of symmetric multiplicative 2-cocycles. This scheme classifies certain kinds of highly symmetric multiextensions, as studied in general by Mumford or Breen. A low-order version of this computation has previously found application in homotopy theory through the sigma-orientation of Ando, Hopkins, and Strickland, and the complete computation is reflective of certain structure found in the homotopy type of connective K-theory. This paper has been completely rewritten from its first posted draft, including a correction of the statement of the main result.