Dipaths and dihomotopies in a cubical complex
Dipaths and dihomotopies in a cubical complex
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立方复形中的二径和二同伦
DOI：
10.1016/j.aam.2005.02.003
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发表时间：
2005
期刊：
影响因子：
--
通讯作者：
L. Fajstrup
中科院分区：
文献类型：
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作者：
L. Fajstrup
在没有简并、□集的立方复形的几何实现中，双径和二同伦可能不是组合的，即不是组合双径和等价的几何实现。当我们想要使用几何/拓扑工具对 1-骨架上的双路径、组合双路径、直至二同伦、特别是组合二同伦进行分类时，我们需要所有双路径实际上都是组合双路径的二同伦。此外，两个双同伦的组合双路径也是组合双同伦的。我们证明，在非自交 □ 集中，从一个顶点到另一个顶点的任何双路径对于组合双路径都是双同伦的。并且通过非组合二同伦而成为二同伦的两个组合二路径实际上在几何□集中是组合二同伦的。此外，我们证明了在几何□集合中，[M. Grandis，有向同伦理论，I，Cah。白杨。杰姆.不同。类别。 44 (4) (2003) 281–316] 与 [L. 44 (4) (2003) 281–316] 中的二同伦一致。 Fajstrup、E. Goubault、M. Raussen，代数拓扑和并发，理论。计算。科学，正在出版；还有技术报告，奥尔堡大学，1999 年]。
In the geometric realization of a cubical complex without degeneracies, a □-set, dipaths and dihomotopies may not be combinatorial, i.e., not geometric realizations of combinatorial dipaths and equivalences. When we want to use geometric/topological tools to classify dipaths on the 1-skeleton, combinatorial dipaths, up to dihomotopy, and in particular up to combinatorial dihomotopy, we need that all dipaths are in fact dihomotopic to a combinatorial dipath. And moreover that two combinatorial dipaths which are dihomotopic are then combinatorially dihomotopic. We prove that any dipath from a vertex to a vertex is dihomotopic to a combinatorial dipath, in a non-selfintersecting □-set. And that two combinatorial dipaths which are dihomotopic through a non-combinatorial dihomotopy are in fact combinatorially dihomotopic, in a geometric □-set. Moreover, we prove that in a geometric □-set, the d-homotopy introduced in [M. Grandis, Directed homotopy theory, I, Cah. Topol. Géom. Différ. Catég. 44 (4) (2003) 281–316] coincides with the dihomotopy in [L. Fajstrup, E. Goubault, M. Raussen, Algebraic topology and concurrency, Theoret. Comput. Sci., in press; also technical report, Aalborg University, 1999].