We construct a space $\mathbb{P}$ for which the canonical homomorphism ${\pi }_1(\mathbb{P},p)\to {\check{\pi }}_1(\mathbb{P},p)$ from the fundamental group to the first Čech homotopy group is not injective, although it has all of the following properties: (1) $\mathbb{P}\setminus \{p\}$ is a 2‐manifold with connected non‐compact boundary; (2) $\mathbb{P}$ is connected and locally path connected; (3) $\mathbb{P}$ is strongly homotopically Hausdorff; (4) $\mathbb{P}$ is homotopically path Hausdorff; (5) $\mathbb{P}$ is 1‐UV ${}_0$; (6) $\mathbb{P}$ admits a simply connected generalized covering space with monodromies between fibers that have discrete graphs; (7) ${\pi }_1(\mathbb{P},p)$ naturally injects into the inverse limit of finitely generated free monoids otherwise associated with the Hawaiian Earring; (8) ${\pi }_1(\mathbb{P},p)$ is locally free.

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关于第一个Čech同伦群未能记录几何相关的基本群元素

我们建造一个空间 $\mathbb{P}$ 其规范同态 ${\pi }_{\mathrm{1个}}（\mathbb{P}，p）\to {\check{\pi }}_{\mathrm{1个}}（\mathbb{P}，p）$ 从基本族到第一个Čech同伦族，虽然具有以下所有特性，但不是内射性的：（1） $\mathbb{P}\setminus \{p\}$是具有连通的非紧边界的2流形; （2）$\mathbb{P}$已连接且本地路径已连接；（3）$\mathbb{P}$与Hausdorff具有强烈的同位性；（4）$\mathbb{P}$是同位路径Hausdorff；（5）$\mathbb{P}$ 是1-UV ${}_0$; （6）$\mathbb{P}$允许一个简单连接的广义覆盖空间，在具有离散图形的光纤之间具有单调性；（7）${\pi }_{\mathrm{1个}}（\mathbb{P}，p）$自然地将有限生成的自由类四面体的倒数注入到否则与夏威夷耳环相关的事物中；（8）${\pi }_{\mathrm{1个}}（\mathbb{P}，p）$ 在本地免费。