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Register a new userOn the failure of the first v{C}ech homotopy group to register geometrically relevant fundamental group elements
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We construct a space $mathbb{P}$ for which the canonical homomorphism $pi_1(mathbb{P},p) rightarrow check{pi}_1(mathbb{P},p)$ 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.
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