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

## TL;DR

This paper constructs a space where the canonical homomorphism from the fundamental group to the cech homotopy group is not injective, despite the space having many properties that typically relate these groups.

## Contribution

It provides a counterexample demonstrating the failure of the first cech homotopy group to detect certain fundamental group elements, challenging assumptions about their relationship.

## Key findings

- The space onstructed has a non-injective canonical homomorphism.
- emonstrates the failure of cech homotopy group to register all fundamental group elements.
- The space exhibits properties like being homotopically Hausdorff and having a simply connected generalized covering space.

## Abstract

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.