Remarks on pointed digital homotopy
Remarks on pointed digital homotopy
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发表时间：
2015-03
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通讯作者：
Laurence Boxer;Christopher Staecker
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作者：
Laurence Boxer;Christopher Staecker
我们提出并详细探讨了一对数字图像的$c_u$-邻接同伦，但不指向同伦。对于具有相同基点的两个数字回路f，g：[0，m] Z \rightarrow X，我们引入了{\em tight at the basepoint（TAB）}指向同伦的概念，它比普通指向同伦的限制性更强，并得到了一些不同的结果.我们提出了一个数字基本群的变体形式。基于我们称之为{\em eventually constant}的循环，这个版本的基本群等价于Boxer（1999）的版本，但提供了一个优点，即最终常数映射通常比Boxer（1999）和许多后续论文中基本群发展的关键的平凡扩展更容易处理。我们证明了同伦等价的数字图像具有同构的基本群，即使当同伦等价不保持基点。这个断言出现在Boxer（2005）中，但是在证明中有一个错误;在这里，我们纠正了这个错误。
We present and explore in detail a pair of digital images with $c_u$-adjacencies that are homotopic but not pointed homotopic. For two digital loops $f,g: [0,m]_Z \rightarrow X$ with the same basepoint, we introduce the notion of {\em tight at the basepoint (TAB)} pointed homotopy, which is more restrictive than ordinary pointed homotopy and yields some different results. We present a variant form of the digital fundamental group. Based on what we call {\em eventually constant} loops, this version of the fundamental group is equivalent to that of Boxer (1999), but offers the advantage that eventually constant maps are often easier to work with than the trivial extensions that are key to the development of the fundamental group in Boxer (1999) and many subsequent papers. We show that homotopy equivalent digital images have isomorphic fundamental groups, even when the homotopy equivalence does not preserve the basepoint. This assertion appeared in Boxer (2005), but there was an error in the proof; here, we correct the error.