Pull-back of metric currents and homological boundedness of BLD-elliptic spaces

Using the duality of metric currents and polylipschitz forms, we show that a BLD-mapping $f\text{\hspace{0.17em}⁣}:X\to Y$ between oriented cohomology manifolds $X$ and $Y$ induces a pull-back operator $f^{\ast }\text{\hspace{0.17em}⁣}:M_{k,loc}(Y)\to M_{k,loc}(X)$ between the spaces of metric $k$-currents of locally finite mass. For proper maps, the pull-back is a right-inverse (up to multiplicity) of the push-forward $f_{\ast }\text{\hspace{0.17em}⁣}:M_{k,loc}(X)\to M_{k,loc}(Y)$. As an application we obtain a non-smooth version of the cohomological boundedness theorem of Bonk and Heinonen for locally Lipschitz contractible cohomology $n$-manifolds $X$ admitting a BLD-mapping ${\mathbb{R}}^n\to X$.