On the failure of the first Čech homotopy group to register geometrically relevant fundamental group elements

Abstract

We construct a space P for which the canonical homomorphism π1(P, p) →π1(P, p) from the fundamental group to the firstČech homotopy group is not injective, although it has all of the following properties: (1) P \ {p} is a 2-manifold with connected non-compact boundary; (2) P is connected and locally path connected; (3) P is strongly homotopically Hausdorff; (4) P is homotopically path Hausdorff; (5) P is 1-UV0; (6) P admits a simply connected generalized covering space with monodromies between fibers that have discrete graphs; (7) π1(P, p) naturally injects into the inverse limit of finitely generated free monoids otherwise associated with the Hawaiian Earring; (8) π1(P, p) is locally free.