Batanin, M. A. Transformations of spectral sequences and the generalized homotopy axiom. (Russian) Zbl 0639.55013 Sib. Mat. Zh. 28, No. 5(165), 22-31 (1987). In this paper the author studies some morphisms of spectral sequences with deplaced degree. By using the developed method the author solves the following problem. Let \(f: M\to K\) be a chain map of complexes with values in an Abelian category inducing the null morphism in homology; T a covariant, additive, left exact functor from the category \({\mathcal A}\) into another Abelian category. The problem is to establish when Tf: TM\(\to TK\) induces the null morphism in homology. In the paper the author obtains a “satisfactory” solution for this problem, imposing some conditions on the category \({\mathcal A}\) and on the chain complexes K and M. The last paragraph of the paper is devoted to an application of this result in a topological problem known as “generalized homotopy axiom” [U. Kh. Karimov, Dokl. Akad. Nauk Tadzh. SSR 22, 521-524 (1979; Zbl 0434.55007) and Nguyen Le Anh, Mat. Zametki 33, No.1, 117-130 (1983; Zbl 0515.55004)]. Concretely, let Y be a paracompact space, X a compact connected space and \(a_ 1,a_ 2\in X\). Consider two imbeddings \(i_ 1\), \(i_ 2: Y\to Y\times X\), \(i_ k(y)=(y,a_ k)\), \(k=1,2\). Demand when \(i_ 1\), \(i_ 2\) induce the same homomorphism in homology. For example, Karimov [loc. cit.] proved when the induced homomorphisms are different. In detail the problem was studied by Nguyen [loc. cit.]. Using an algebraic approach the author obtains some refinement of the results given by Nguyen. Reviewer: Ioan Pop (Iaşi) Cited in 1 ReviewCited in 2 Documents MSC: 55U15 Chain complexes in algebraic topology 55N40 Axioms for homology theory and uniqueness theorems in algebraic topology 55N99 Homology and cohomology theories in algebraic topology Keywords:induced homomorphisms in homology; chain map of complexes with values in an Abelian category inducing the null morphism in homology; covariant, additive, left exact functor; generalized homotopy axiom; paracompact space; compact connected space; imbeddings Citations:Zbl 0434.55007; Zbl 0515.55004 PDFBibTeX XMLCite \textit{M. A. Batanin}, Sib. Mat. Zh. 28, No. 5(165), 22--31 (1987; Zbl 0639.55013) Full Text: EuDML