Unlike Čech cohomology, Čech homology is not a homology theory in the sense of Eilenberg-Steenrod: the exactness axiom (long exact sequence in homology) does not hold. A correction to the basic Čech definition was given by Sibe Mardešić. The resulting “strong homology theory” agrees with singular homology on the spaces having homotopy type of CW complexes, and does give long exact sequence of pairs $(X,A)$ where $X$ is paracompact and $A$ closed in $X$; moreover for metric compacta it satisfies not only all the axioms of Eilenberg-Steenrod, but also the relative homeomorphism axiom and the wedge axiom.

The only homology theory on the metric compacta satisfying not only the Eilenberg-Steenrod but also the wedge axiom is the Steenrod-Sitnikov homology? theory, hence the strong homology agrees with it.

References

S. Mardešić, Strong shape and homology, Springer monographs in mathematics, 2000. xii+489 pp.

Revised on October 6, 2010 17:33:17
by Tim Porter
(95.147.237.88)