Projective and special homologies
type: Item
instance of: scholarly article
title: Projective and special homologies
zbMATH Open document ID: 1082.55002
author: Sh. Bakhtadze
published in: Journal of Mathematical Sciences (New York)
publication date: 2005-06-07T00:00:00Z
review text: 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).)
reviewed by: Gábor Braun
Mathematics Subject Classification ID: 55N05
Mathematics Subject Classification ID: 55N07
Mathematics Subject Classification ID: 55N40
zbMATH DE Number: 2174277
zbMATH Keywords: projective homology
zbMATH Keywords: special homology
zbMATH Keywords: Chogoshvili homology
zbMATH Keywords: Steenrod homology
MaRDI profile type: Publication
DOI: 10.1023/B:JOTH.0000009371.43936.5A
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description: scientific article; zbMATH DE number 2174277