Geodesic
Spaces of differential forms and maps with controlled distortion
Izvestiya. Mathematics , Tome 74 (2010) no. 4, pp. 663-689

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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.
Keywords: Lebesgue space of differential forms, distortion of a map, quasi-conformal mapping, cohomology of Riemannian spaces. 
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     author = {S. K. Vodop'yanov},
     title = {Spaces of differential forms and maps with controlled distortion},
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     year = {2010},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2010_74_4_a1/}
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S. K. Vodop'yanov. Spaces of differential forms and maps with controlled distortion. Izvestiya. Mathematics , Tome 74 (2010) no. 4, pp. 663-689. http://geodesic.mathdoc.fr/item/IM2_2010_74_4_a1/