Nonlocal loss of first homotopy in polyhedral approximations of Peano continua

TL;DR

This paper investigates the limitations of polyhedral approximations in capturing the fundamental group of Peano continua, revealing that certain nonlocal properties cause persistent non-detectability of elements.

Contribution

It demonstrates that the nonlocal nature of fundamental group failures in polyhedral approximations is intrinsic, extending understanding beyond local properties.

Findings

Polyhedral approximations can fail to detect some fundamental group elements.

Failure of approximation is not solely due to local properties.

Nonlocal phenomena cause persistent undetectability in approximations.

Abstract

If a Peano continuum $X$ is semilocally simply connected, then it has a finite polyhedral approximation whose fundamental group is isomorphic to that of $X$. In general, this fails to be true. It is known that the fundamental group of a locally complicated Peano continuum may contain nontrivial elements that are persistently undetectable by polyhedral approximations, at all scales. However, we show that such failure is not inherently local.