Pull-Back of Metric Currents and Homological Boundedness of BLD-Elliptic Spaces

Abstract

Using the duality of metric currents and polylipschitz forms, we show that a BLD-mappingf : X -> Y between oriented cohomology manifolds X and Y induces a pull-back operator f * : M-k,M-loc(Y) -> M-k,M-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(*): M-k,M-loc(X) -> M-k,M-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 R-n -> X.