The realizability of operations on homotopy groups concentrated in two degrees

TL;DR

This paper investigates the conditions under which homotopy operations can realize a-algebras concentrated in two degrees, extending known results from the one-degree case and identifying non-realizable cases in the stable setting.

Contribution

It provides necessary and sufficient conditions for realizing a-algebras in two degrees and characterizes non-realizable cases in the stable context.

Findings

Identifies conditions for a-algebra realizability in two degrees

Lists infinite families of non-realizable a-algebras in the stable case

Extends the understanding of homotopy operations beyond the one-degree case

Abstract

The homotopy groups of a space are endowed with homotopy operations which define the \Pi-algebra of the space. An Eilenberg-MacLane space is the realization of a \Pi-algebra concentrated in one degree. In this paper, we provide necessary and sufficient conditions for the realizability of a \Pi-algebra concentrated in two degrees. We then specialize to the stable case, and list infinite families of such \Pi-algebras that are not realizable.