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.
Publié le : 2007-04-14
Classification:
Čech homology,
singular homology,
locally connected spaces,
integral flat chains,
integral currents,
49Q15,
55N35,
55N40,
58A25
@article{1180728888,
author = {De Pauw, Thierry},
title = {Comparing homologies: \v Cech's theory, singular chains, integral flat chains and integral currents},
journal = {Rev. Mat. Iberoamericana},
volume = {23},
number = {1},
year = {2007},
pages = { 143-189},
language = {en},
url = {http://dml.mathdoc.fr/item/1180728888}
}
De Pauw, Thierry. Comparing homologies: Čech's theory, singular chains, integral flat chains and integral currents. Rev. Mat. Iberoamericana, Tome 23 (2007) no. 1, pp. 143-189. http://gdmltest.u-ga.fr/item/1180728888/