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We prove that every (geometrically) quasiconformal homeomorphism betweenmetric measure spaces induces an isomorphism between the cotangent modulesconstructed by Gigli. We obtain this by first showing that every continuousmapping $\varphi$ with bounded outer dilatation induces a pullback map$\varphi^*$ between the cotangent modules of Gigli, and then proving thefunctorial nature of the resulting pullback operator. Such pullback isconsistent with the differential for metric-valued locally Sobolev mapsintroduced by Gigli-Pasqualetto-Soultanis.  Using the consistency between Gigli's and Cheeger's cotangent modules for PIspaces, we prove that quasiconformal homeomorphisms between PI spaces preservethe dimension of Cheeger charts, thereby generalizing earlier work byHeinonen-Koskela-Shanmugalingam-Tyson.  Finally, we show that if $\varphi$ is a given homeomorphism with boundedouter dilatation, then $\varphi^{-1}$ has bounded outer dilatation if and onlyif $\varphi^{*}$ is invertible and $\varphi^{-1}$ is Sobolev. In contrast tothe setting of Euclidean spaces, Carnot groups, or more generally, Ahlforsregular PI spaces, the Sobolev regularity of $\varphi^{-1}$ needs to be assumedseparately.

Pullback of a quasiconformal map between arbitrary metric measure spaces