Nonlocal loss of first homotopy in polyhedral approximations of Peano continua

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.

23 pages, 11 figures

References in corpus (3)

• On small homotopies of loops
• The Griffiths double cone group is isomorphic to the triple
• A simply connected universal fibration with unique path lifting over a Peano continuum with non-simply connected universal covering space