Projective and special homologies (Q1780365)

Projective homology is a variant of Čech homology, i.e.\ a homology theory for pairs of compact Hausdorff spaces. It is a homology theory based on simplical complexes associated to partitions. The author claims to cover the basics of projective and special homology theory in Sections 1.1 to 1.3 including their equivalence. Sections 1.4 and 1.5 define these homologies. Section 1.6 claims to prove that projective homology is a generalization of Steenrod--Sitnikov homology groups from \textit{N. E. Steenrod}, [Ann. Math. (2) 41, 833--851 (1940; Zbl 0025.23405)]. Section 2 claims to prove that projective homology satisfies the usual Eilenberg-Steenrod axioms of homology theory and Milnor's axioms on infinite wedges and relative homeomorphisms. Finally, Section 3 claims to modify the theory to obtain extraordinary homology theories. There are significant mathematical inaccuracies in the paper. For example, a ``piece of chain'', first appearing just before Definition 1.3.1, is not defined, the reader has to figure it out himself. In the second sentence of the proof of Theorem 1.3.1, the phrase ``a straightforward passage to quotient groups'' uses 1.3.2, which is a corollary to the theorem. The statement of Lemma 1.6.5 is false because no complex has a proper subcomplex such that every element of the complex is the sum of a boundary and an element of a subcomplex. (The proof does not show condition (c), and the proof of condition (b) tacitly assumes (wrongly) that \(x_p\) satisfies condition (b).)