  We study PL bordism theories from a quantitative perspective. Such theories
include those of PL manifolds, ordinary homology theory, as well as various
more exotic theories such as bordism of Witt spaces. In all these cases we show
that a null-bordant cycle of bounded geometry and $V$ simplices has a filling
of bounded geometry whose number of simplices is slightly superlinear in $V$.
This bound is similar to that found in our previous work on smooth cobordism.
