F. W. Bauer

Strong Homology Theories as Localizations

Let ${\mathfrak K}$ be a category of pairs of spaces, ${\mathfrak L} \subset {\mathfrak K}$ the category of pairs of ANRs or CW-spaces, $A_*$ a chain functor (e.g., one associated with a spectrum). Then the derived homology ${}^s h_*$ of the ${\mathfrak L}$-localization of $A_*$ is the strong homology theory on ${\mathfrak K}$ which is up to an isomorphism uniquely determined by the fact that ${}^sh_* \mid {\mathfrak L}$ agrees with the derived homology of $A_*$ on ${\mathfrak L}$. This establishes a relationship between localization theory and strong homology theory (e.g., Steenrod-Sitnikov homology theory, whenever all pairs are compact metric).

Localizations, strong homology theory, chain functors..

MSC 2000: Primary: 55P60, 55N07, 55N99; secondary:  55N40, 55N20, 55U15