On the splitting of polynomial functors

TL;DR

This paper develops methods to determine when extensions of polynomial functors do not split and applies these to describe certain stable homotopy groups of classifying spaces.

Contribution

It introduces new techniques for analyzing non-splitting of polynomial functor extensions and provides a functorial description of specific stable homotopy groups.

Findings

Extensions of polynomial functors can be shown not to split naturally.

A functorial description of the third and fourth stable homotopy groups of classifying spaces of free abelian groups.

New methods for studying the structure of polynomial functors and their extensions.

Abstract

We develop methods for proving that certain extensions of polynomial functors do not split naturally. As an application we give a functorial description of the third and the fourth stable homotopy groups of the classifying spaces of free abelian groups.