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

TL;DR

This paper establishes a duality-based framework for pull-back operations on metric currents induced by BLD-mappings between cohomology manifolds, extending classical cohomological boundedness results to non-smooth settings.

Contribution

It introduces a duality approach linking metric currents and polylipschitz forms to define pull-back operators for BLD-maps, generalizing cohomological boundedness to non-smooth manifolds.

Findings

Pull-back operator is a right-inverse of push-forward for proper maps.

Extension of cohomological boundedness theorem to non-smooth BLD-elliptic spaces.

Framework applies to locally Lipschitz contractible cohomology manifolds.

Abstract

Using the duality of metric currents and polylipschitz forms, we show that a BLD-mapping $f\colon X\to Y$ between oriented cohomology manifolds $X$ and $Y$ induces a pull-back operator $f^\ast \colon M_{k,loc}(Y) \to M_{k,loc}(X)$ 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 $f_\ast \colon M_{k,loc}(X)\to M_{k,loc}(Y)$. 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 $\mathbb{R}^n \to X$.