Comparing homologies: Čech's theory, singular chains, integral flat chains and integral currents

We give a new proof of a Theorem of S. Mardesic, generalized by G.E. Bredon, that Cech 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 Cech's. Relations between these theories is relevant for the solvability and regularity of many geometric variational problems.