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-mapping f : X → Y between oriented cohomology manifolds X and Y induces a pull-back operator f * : M k,loc (Y ) → 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 * : M k,loc (X) → 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 R n → X.Date: February 20, 2019. 2010 Mathematics Subject Classification. 30L10, 49Q15, 30C65. Key words and phrases. Metric currents, BLD-mappings. P.P. was supported in part by the Academy of Finland project #297258.

Metric currents and polylipschitz forms

“…Our motivation to consider polylipschitz forms stems from an application of metric currents to geometric mapping theory -polylipschitz forms induce a natural local pull-back for metric currents of finite mass under BLD-mappings. We discuss this application briefly in the end of the introduction and in more detail in [9].…”

“…Then there exists a unique sequentially continuous linear functional Motivation: Pull-back of metric currents by BLD-maps. In [9] we apply the duality theory developed in this paper to a problem in geometric mapping theory. To avoid the added layer of abstraction involved in polylipschitz forms we formulate the results in [9] for polylipschitz sections, which are sufficient for our purposes.…”

“…In [9] we apply the duality theory developed in this paper to a problem in geometric mapping theory. To avoid the added layer of abstraction involved in polylipschitz forms we formulate the results in [9] for polylipschitz sections, which are sufficient for our purposes. For this reason, in Section 9 we briefly discuss duality theory in connection with polylipschitz sections.…”

“…In the classical setting of Euclidean spaces, this push-forward is associated to the pull-back of differential forms under the mapping f . In [9], we consider BLD-mappings f : X → Y between metric generalized n-manifolds. A mapping f : X → Y is a mapping of bounded length distortion (or BLD for short) if f is a discrete and open mapping for which there exists a constant…”

“…For a BLD-mapping f : X → Y between locally compact spaces, the polylipschitz sections admit a push-forward f # : Γ k c (X) → Γ k pc,c (Y ), which in turn induces a natural pull-back f * : M k,loc (Y ) → M k,loc (X) for metric currents. We refer to [9] for detailed statements and further applications.…”