Category and Group Rings in Homotopy Theory

Abstract

It frequently arises in algebraic topology that a function $\beta :G \to H$, between two groups, is not a homomorphism. We show that in many standard situations $\beta$ induces a group homomorphism $\overline \beta :{\mathbf {Z}}(G)/{\mathcal {A}^d} \to H$, where ${\mathcal {A}^d}$ is a power of the augumentation ideal in the group ring ${\mathbf {Z}}(G)$. A typical example is $\beta :[X, Y] \to [{S^2}X, {S^2}Y]$ where $Y$ is some $H$-group, in which case $d$ can be taken to be $1 + {\text {cat}} X$.