Spaces of differential forms and maps with controlled distortion

We study necessary and sufficient conditions for an approximately differentiable map $f\colon\mathbb M\to\mathbb M'$ between Riemannian manifolds to induce a bounded transfer operator of differential forms with respect to the norms of Lebesgue spaces. As a corollary, we see that every homeomorphism $f\colon\mathbb M\to\mathbb M'$ of class $\operatorname{ACL}(\mathbb M)$ whose transfer operator of differential forms with norm in $\mathcal L_p$ is an isomorphism must necessarily be either quasi-conformal or quasi-isometric. We give some applications of our results to the study of the functoriality of cohomology in Lebesgue spaces.