**F. W. Bauer**

## Strong Homology Theories as Localizations

**Abstract:**

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).

**Keywords:**

Localizations, strong homology theory, chain functors..

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