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On the failure of the first Čech homotopy group to register geometrically relevant fundamental group elements
Bulletin of the London Mathematical Society ( IF 0.787 ) Pub Date : 2020-06-28 , DOI: 10.1112/blms.12383
Jeremy Brazas; Hanspeter Fischer

We construct a space P for which the canonical homomorphism π 1 ( P , p ) → π ̌ 1 ( 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; (8) π 1 ( P , p ) is locally free.
更新日期:2020-06-29

 

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