Revista Matemática Iberoamericana
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Published online: 2007-04-30
Comparing homologies: Čech's theory, singular chains, integral flat chains and integral currentsPierre Fima (1) Université Denis Diderot (Paris 7), France
We give a new proof of a Theorem of S. Mardešić, generalized by G. E. Bredon, that Čech and singular homology groups of certain locally connected spaces coincide. We use the chain complexes of integral flat chains (H. Whitney) and integral currents (H. Federer and W. H. Fleming) to define new homology groups of subsets of Euclidean space. We show these verify the axioms of Eilenberg and Steenrod, and we provide Lipschitz-flavored local connectedness conditions which guarantee these groups coincide with Čech's. Relations between these theories is relevant for the solvability and regularity of many geometric variational problems.
Keywords: Čech homology, singular homology, locally connected spaces, integral flat chains, integral currents
Fima Pierre: Comparing homologies: Čech's theory, singular chains, integral flat chains and integral currents. Rev. Mat. Iberoam. 23 (2007), 143-189. doi: 10.4171/RMI/489