nLab strong homology

Strong homology theory is a homology theory of topological spaces which in addition to satisfying Eilenberg-Steenrod axioms is an invariant of strong shape. It has been developed in a series of papers by Lisica and Mardešić.

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 (“additivity”). However this axiom is not satisfied for general spaces; the counterexamples have been found by Prasolov and Mardešić.

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

A comprehensive reference is

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

Sensitivity of the results on vanishing of (higher) derived limit functors on additional axioms of set theory, and consequences on strong homology are discussed in

• Jeffrey Bergfalk, Strong homology, derived limits, and set theory, arXiv/1509.09267

Extraordinary strong homology theories (represented by usual spectra; in the case of Eilenberg-MacLane spectrum agree with the usual strong homology) are constructed in

• Andrei Prasolov, Extraordinary strong homology, Topology and its Applications 113(1):249-291 (2001)