The third homotopy module of a 2-complex

TL;DR

This paper develops methods to compute the third homotopy group of a 2-complex as a module over the group ring, and shows how its stable class is determined by the fundamental group when finite of odd order.

Contribution

It provides a new approach to compute pi_3 of 2-complexes as modules over Z[G] and characterizes their stable class for finite groups of odd order.

Findings

pi_3(X) can be computed as a Z[G]-module

The stable class of pi_3(X) is determined by G for finite groups of odd order

Provides explicit methods for 2-complexes with finite fundamental groups

Abstract

Given a connected 2-complex X with fundamental group G, we show how pi_3(X) may be computed as a module over Z[G]. Further we show that if X is a finite connected 2-complex with G (the fundamental group) finite of odd order, then the stable class of pi_3(X) is determined by G.