A classification of additive symmetric 2-cocycles

Abstract

We present a classification of the so-called “additive symmetric 2-cocycles” of arbitrary degree and dimension over $\mathbb{F}_p$, along with a partial result and some conjectures for $m$-cocycles over $\mathbb{F}_p$, $m \gt 2$. This expands greatly on a result originally due to Lazard and more recently investigated by Ando, Hopkins and Strickland, and together with their work this culminates in a complete classification of $2$-cocycles over an arbitrary commutative ring. The ring classifying these polynomials finds application in algebraic topology, to be fully explored in a sequel.