On the failure of the first \v{C}ech homotopy group to register geometrically relevant fundamental group elements

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 \v{C}ech 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.