Free and non-free subgroups of the fundamental group of the Hawaiian Earrings
Free and non-free subgroups of the fundamental group of the Hawaiian Earrings
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夏威夷耳环基本群的自由和非自由子群
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发表时间：
2003
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通讯作者：
A. Zastrow
中科院分区：
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作者：
A. Zastrow
空间是由嵌入可数多个圆在这样一种方式到平面上，他们的半径是由一个零序列，他们都有一个共同的切点被称为“夏威夷耳环”。这个空间的基本群是已知的逆极限自由群的子群，并且它是不自由的。在最近的努力中，试图得到非驯服（例如不可三角化）空间的代数不变量，这个空间通常是这方面最简单的例子。本文有助于理解这个群和相应的现象，指出几个子群，根据类似的计划，部分地被证明是自由的，而不是自由的。其中有一个可数的非自由子群和一个不可数的自由子群，这个自由子群不包含在最近发现的另外两个自由子群中。这个群虽然是自由的，但包含无限大的“虚幂”，即通常用于证明这个基本群不自由的基本群的元素，并且，虽然这个群包含了与夏威夷耳环的单个环相关联的所有同伦类的路径，但是可以证明这个“自然生成元”系统不包含在这个自由群的任何自由基中。
The space which is composed by embedding countably many circles in such a way into the plane that their radii are given by a null-sequence and that they all have a common tangent point is called “The Hawaiian Earrings”. The fundamental group of this space is known to be a subgroup of the inverse limit of the finitely generated free groups, and it is known to be not free. Within the recent move of trying to get hands on the algebraic invariants of non-tame (e.g. non-triangulable) spaces this space usually serves as the simplest example in this context. This paper contributes to understanding this group and corresponding phenomena by pointing out that several subgroups that are constructed according to similar schemes partially turn out to be free and not to be free. Amongst them is a countable non-free subgroup, and an uncountable free subgroup that is not contained in two other free subgroups that have recently been found. This group, although free, contains infinitely huge “virtual powers”, i.e. elements of the fundamental group of that kind that are usually used in proofs that this fundamental group is not free, and, although this group contains all homotopy classes of paths that are associated with a single loop of the Hawaiian Earrings, this system of ‘natural generators’ can be proven to be not contained in any free basis of this free group.