Strong homology of compact spaces
This section is devoted to strong homology of compact Hausdorff spaces. In subsection 21.1 we recall the universal coefficient theorem for compact polyhedra. In 21.2 we establish the all-important fact that the higher derived limits lim r H m (X;G) of an inverse system X of compact polyhedra vanish, for r ≥ 2 (Theorem 21.6). It is the proof of this fact that requires the machinery developed in 20. We also prove the Milnor exact sequence (Theorem 21.9). In 21.3 we prove the universal coefficient theorem for compact Hausdorff spaces (Theorem 21.15). In view of the axiomatic characterization of strong homology (Berikashvili 1984), our strong homology of compact Hausdorff spaces coincides with that of several other authors. In 21.4 we obtain a large commutative diagram (see (21.4.1)), which embodies the just mentioned results and contains additional information (Theorem 21.18). The last subsection is devoted to strong homology with compact supports.
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