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April, 2007 Comparing homologies: Čech's theory, singular chains, integral flat chains and integral currents
Thierry De Pauw
Rev. Mat. Iberoamericana 23(1): 143-189 (April, 2007).

Abstract

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.

Citation

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Thierry De Pauw. "Comparing homologies: Čech's theory, singular chains, integral flat chains and integral currents." Rev. Mat. Iberoamericana 23 (1) 143 - 189, April, 2007.

Information

Published: April, 2007
First available in Project Euclid: 1 June 2007

zbMATH: 1246.49038
MathSciNet: MR2351129

Subjects:
Primary: 49Q15, 55N35, 55N40, 58A25

Rights: Copyright © 2007 Departamento de Matemáticas, Universidad Autónoma de Madrid

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Vol.23 • No. 1 • April, 2007
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