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The relative homeomorphism and wedge axioms for strong homology. (English) Zbl 0794.55004

The authors deal with strong homology theories \(\overline{H}_ *\) in the sense of J. T. Lisica and S. Mardešić [Bull. Am. Math. Soc., New Ser. 9, 207–210 (1983; Zbl 0532.55003)], which is always an ordinary homology theory being defined by means of ANR resolutions resp. by polyhedral resolutions of pairs of spaces.
The objective of the present paper is to establish a proof of
(1) A general excision property (the projection \(p: (X,A)\to (X/A,*)\), \(A\neq\emptyset\), induces an isomorphism of homology groups.
(2) A general strong wedge axiom (for a collection \(X_ \iota= (X_ \iota, x_{0\iota})\) of spaces, one has the cluster \(\text{Cl}_{\iota\in J} X_ \iota= X= \varprojlim X_{\iota_ 1}\vee \cdots\vee X_{\iota_ n}\) and the projections \(p_ \iota: X\to X_ \iota\) induce an isomorphism \(\overline{H}_ * (X) \approx \prod_{\iota\in J} \overline{H}_ * (X_ \iota)\)).
For compact metric spaces and accordingly \(J= \{1,2,\dots\}\) (1) and (2) are parts of the Milnor axioms for a Steenrod-Sitnikov homology theory. In particular (2) exhibits a weak form of continuity of the given homology theory. For special classes of pairs (e.g. \(X\) paracompact, \(A\subset X\) closed) these assertions have already been accomplished by T. Watanabe [Glas. Math., III. Ser. 22(42), No. 1, 187–238 (1987; Zbl 0658.55003)]. The authors emphasize that they are able to deal with polyhedral rather than with ANR resolutions which enables them to achieve (1) and (2) for arbitrary spaces.
Reviewer’s remark: For generalized (i.e. not necessarily ordinary) axiomatically defined strong homology theories (2) was settled in the affirmative by the reviewer [Note Mat. 10, Suppl. No. 1, 73–102 (1990; Zbl 0761.55003)]. Here (1) is part of the axioms; another axiom is a continuity property (this time on the chain level). The following questions are still (at least according to may knowledge) open:
I. For what classes of pairs are (1), (2) sufficient to characterize Lisica-Mardešić strong homology theory?
II. Under what conditions is the axiomatically defined strong homology theory isomorphic to the Lisica-Mardešić version (whenever both are defined, e.g. for ordinary homology theories)?

MSC:

55N07 Steenrod-Sitnikov homologies
55N40 Axioms for homology theory and uniqueness theorems in algebraic topology
55P55 Shape theory
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