Generalized homology theories and chain complexes (Q910047)

A full subcategory \(K\subset Top\) is called a category of topological spaces if K satisfies the following axioms: K1) \(\emptyset\), \(\{\) point\(\}\in K\); K2) \(X\in K\) \(\Rightarrow\) \(X\times [0,1]\in K\); K3) If \(X=X_ 1\cup X_ 2\), \(X_ i\subset X\) closed, \(X,X_ i\in K\), then \(X_ 1\cap X_ 2\in K\); K4) If (i: \(A\subset X)\in K\) is a closed inclusion and if \(f\in K(A,Y)\) continuous, then \(X\cup_ fY\in K.\) Let K be a category of topological spaces and \(h_ n: K^ 2\to Ab\), \(\partial_ n: h_ n(X,A)\to h_{n-1}(A)\), \(n\in {\mathbb{Z}}\), be families of functors and natural transformations, respectively, satisfying H1) a homotopy axiom, and H2) an exactness axiom, then \(h=\{h_ n,\partial_ n\}\) is called a prehomology theory \((K^ 2\) is the category of pairs). There are two different excision axioms, namely: H3) Ordinary excision axiom: Let \((X,A)\in K^ 2\), \(\bar U\subset Int A\), U open, \((X\setminus U,A\setminus U)\in K^ 2\), then the inclusion induces an isomorphism \(h_ n(X\setminus U,A\setminus U)\tilde{\tilde h}_ n(X,A)\), \(n\in {\mathbb{Z}}.\) H3\({}') \) Strong excision axiom: Let \((X,A)\in K^ 2\) be a pair such that \((X/A,*)\in K^ 2\), then the projection p: (X,A)\(\to (X/A,*)\) induces an isomorphism \(h_ n(p): h_ n(X,A)\to h_ n(X/A,*).\) A chain functor \(C_*: K\to ch\) \((=category\) of chain complexes) is a functor which satisfies four special axioms (Definition 2.1). A chain functor \(C_*: K\to ch\) and a prehomology theory \(h_*: K^ 2\to Ab^{{\mathbb{Z}}}\), \(h_*=\{h_ n,\partial_ n\}\) are defined to be ``related to each other'' whenever there exists a natural isomorphism of prehomology theories \(\mu\) : \(H_*(C_*)\to h_*( )\). In this paper to each generalized homology theory \(h_*\), defined on a category of topological spaces K, a chain functor \(C_*: K\to ch\) is established, which is related to \(h_*\) (Theorem 8.1). The author announces that in subsequent papers this result is used for constructing a strong homology theory (i.e. an analogue of the Steenrod-Sitnikov homology theory for general topological spaces [cf. \textit{F. W. Bauer}, Lect. Notes Math. 1283, 3-29 (1987; Zbl 0631.55004)].