 Allowable singular simplex

An $i$-dimensional singular simplex $\sigma :{\Delta }^i\to X$ is $\bar{p}$-allowable if ${\sigma }^{-1}(X_{N-k})$ is contained in the $(i-k+\bar{p}(k))$-skeleton of ${\Delta }^i$ for every $k\ge 2$. A negative skeleton is empty. Individual faces need not be allowable; this is why the definition of intersection chains separately imposes allowability on the total boundary.