Quantitative PL bordism

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.