Čech type approach to computing homology of maps. (English) Zbl 1219.55004
The article is devoted to the new types of algorithms dealing with computing homology of spaces and maps as well as to the research of various properties. The approach the author presents is based on a combinatorial version of a Čech homology and the Nerve theorem.
The main result of the article is the algorithm for computing the homology of continuous maps. The Mayer-Vietoris theorem for Čech structures is thoroughly presented. The author constructs the chain map between the chain complex of a Čech structure and the singular chain complex. The author shows that the homologies of all Čech structures of a given Čech polyhedron are the so-called connected simple system. We present some of major results.
Theorem 4.1. Let \(\chi\) be a Čech structure. There is a well defined chain map \(\varphi^\chi:C_\#(\chi)\to C_\#(|\chi|)\) which induces an isomorphism of the homology \(H_*(|\chi|)\) of the Čech structure \(\chi\) and the singular homology \(H_*(|\chi|)\) of the support of \(\chi\).
Theorem 11.3. Let \(\chi\) be a Čech structure. Then the map \(\varphi^\chi:C_\#(\chi)\to C_\#(|\chi|)\) induces an isomorphism in homology.
The main result of the article is the algorithm for computing the homology of continuous maps. The Mayer-Vietoris theorem for Čech structures is thoroughly presented. The author constructs the chain map between the chain complex of a Čech structure and the singular chain complex. The author shows that the homologies of all Čech structures of a given Čech polyhedron are the so-called connected simple system. We present some of major results.
Theorem 4.1. Let \(\chi\) be a Čech structure. There is a well defined chain map \(\varphi^\chi:C_\#(\chi)\to C_\#(|\chi|)\) which induces an isomorphism of the homology \(H_*(|\chi|)\) of the Čech structure \(\chi\) and the singular homology \(H_*(|\chi|)\) of the support of \(\chi\).
Theorem 11.3. Let \(\chi\) be a Čech structure. Then the map \(\varphi^\chi:C_\#(\chi)\to C_\#(|\chi|)\) induces an isomorphism in homology.
Reviewer: Andreiy Kondrat’yev (Birmingham/AL)
MSC:
| 55N99 | Homology and cohomology theories in algebraic topology |
| 68W30 | Symbolic computation and algebraic computation |
| 55N05 | Čech types |
| 55N10 | Singular homology and cohomology theory |
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