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Transformations of spectral sequences and the generalized homotopy axiom. (Russian) Zbl 0639.55013

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)

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
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