Mardešić, S.; Miminoshvili, Z. The relative homeomorphism and wedge axioms for strong homology. (English) Zbl 0794.55004 Glas. Mat., III. Ser. 25(45), No. 2, 387-416 (1990). 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)? Reviewer: F.W.Bauer (Frankfurt / Main) Cited in 4 Documents MSC: 55N07 Steenrod-Sitnikov homologies 55N40 Axioms for homology theory and uniqueness theorems in algebraic topology 55P55 Shape theory Keywords:strong homology theories; excision; strong wedge axiom; Milnor axioms Citations:Zbl 0532.55003; Zbl 0658.55003; Zbl 0761.55003 PDFBibTeX XMLCite \textit{S. Mardešić} and \textit{Z. Miminoshvili}, Glas. Mat., III. Ser. 25(45), No. 2, 387--416 (1990; Zbl 0794.55004)