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 π 1 (P, p) → π1 (P, p) from the fundamental group to the first Čech 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-UV 0 ; (6) P admits a simply connected generalized covering space with monodromies between fibers that have discrete graphs; (7) π 1 (P, p) naturally injects into the inverse limit of finitely generated free monoids otherwise associated with the Hawaiian Earring.

Scattered products in fundamental groupoids

“…Additionally, many other examples exhibiting more intricate infinite π 1 -product operations are homotopically Hausdorff, e.g. A, B ⊂ R 3 in [9], the space RX in [24], and the "Hawaiian pants" space P ⊂ R 3 in [3]. We refer to [2,9,17] for characterizations and comparisons of the homotopically Hausdorff property with other local properties.…”

“…According to classical covering space theory [26], if X is path connected and locally path connected, then w(X) is the topological obstruction to the existence of a simply connected covering space over X. This set plays a key role in automatic continuity and homotopy classification results for one-dimensional [14,15] and planar [9,23] continua since the homeomorphism type of the topological 1-wild set is a homotopy invariant within these classes of spaces [4,Theorem 9.13].…”