The deformation theorem for flat chains
The author extends the applicability of the deformation theorem of \textit{H. Federer} and \textit{W. H. Fleming} [Ann. Math. (2) 72, 458--520 (1960; Zbl 0187.31301)] to arbitrary flat chains over a complete normed abelian group. The definition of flat chains over coefficient groups other than the integers or the reals cannot be made via differential forms, but rather must proceed along the lines used by \textit{W. H. Fleming} [Trans. Am. Math. Soc. 121, 160--186 (1966; Zbl 0136.03602)]: namely, the space of flat chains is the completion, with respect to the flat metric, of the space of polyhedral chains. The author is able to use his results to answer a number of questions about flat chains; among these results is the fact that, for an arbitrary coefficient group, a nonzero flat \(k\)-chain cannot be supported in a set of \(k\)-dimensional measure zero. In another paper [Ann. Math. (2) 150, No. 1, 165--184 (1999; Zbl 0965.49024)], the author uses the deformation theorem from the paper under review to give a simple necessary and sufficient condition on a coefficient group \(G\) in order for every finite-mass flat chain to be rectifiable.
scientific article; zbMATH DE number 1541219
scientific article