In Mosher & Tangora's book Cohomology Operations and Operations in Homotopy Theory, they define an $h$-equivariant carrier as in the excerpt below. But I don't quite understand the definitions here. $\pi$ is a group,$K$ is a $\pi$-free chain complex with a $Z[\pi]$ basis $B$ of homogeneous elements. I wonder what does a $Z[\pi]$-basis means here for a chain complex.
To state the fundamental theorem on acyclic carriers, we need some terminology. Let $\pi$ and $G$ be groups (not necessarily abelian) and let $\mathbb{Z}[\pi]$ denote the group ring of $\pi$. Let $K$ be a $\pi$-free chain complex with a $\mathbb{Z}[\pi]$-basis $B$ of homogeneous elements, called "cells." For two cells $\sigma, \tau \in B$, let $[\tau : \sigma]$ denote the coefficient of $\sigma$ in $\partial \tau$; this is an element in $\mathbb{Z}[\pi]$. Let $L$ be a chain complex on which $G$ acts, and let $h$ be a homomorphism $\pi \to G$.
Definition: An $h$-equivariant carrier $\mathcal{C}$ from $K$ to $L$ is a function $\mathcal{C}$ from $B$ to the subcomplexes of $L$ such that:
- if $[\tau : \sigma] \neq 0$ then $\mathcal{C} \sigma \subset \mathcal{C} \tau$
- for $x \in \pi$ and $\sigma \in B$, $h(x) \mathcal{C} \sigma \subset \mathcal{C} \sigma$
The carrier $\mathcal{C}$ is said to be acyclic if the subcomplex $\mathcal{C} \sigma$ is acyclic for every cell $\sigma \in B$. The $h$-chain map $f : K \to L$ is said to be carried by $\mathcal{C}$ if $f \sigma \in \sigma \mathcal{C}$ for every $\sigma \in B$.
