Generalized Steenrod homology theories are strong shape invariant (Q985323)

The notion of Steenrod homology with respect to strong shape theory is studied in this paper. The former has been developed both in the unpointed and pointed cases (references [\textit{J. Kaminker} and \textit{C. Schochet}, Trans. Am. Math. Soc. 227, 63--107 (1977; Zbl 0368.46054)] and [\textit{D. S. Kahn, J. Kaminker} and \textit{C. Schochet}, Mich. Math. J. 24, 203--224 (1977; Zbl 0384.55001)] of the paper are suggested for background). The author focuses on the pointed case, and in particular, what is called the reduced theory. The topic of strong shape theory is covered in [\textit{S. Mardešić}, Strong shape and homology. Springer Monographs in Mathematics. Berlin: Springer. (2000; Zbl 0939.55007)]. It is shown that all generalized Steenrod homology theories are strong shape invariant. Some important results of the paper occur in Section 6. It is shown that strong shape invariant reduced pointed homology theories, strong shape invariant reduced unpointed homology theories, and strong shape invariant unreduced homology theories are equivalent. This means that any one can be transformed into another without losing information. In Section 7, the author generalizes the fact that in the absolute case, the strong shape functor on the category \(\mathbf{CM}\) of metrizable compacta localizes \(\mathbf{CM}\) at the class of strong shape equivalences. Herein, the pointed case of this result, Theorem 7.7, is proved.