Homomorphims from the fundamental group of planar sets

Abstract

We prove that every homomorphism from the fundamental group of a planar Peano continuum to the fundamental group of a planar or one-dimensional Peano continuum is induced by a continuous map up to conjugation. This is then used to provide a family of uncountable many planar Peano continua with pairwise non-isomorphic fundamental groups all of which are not homotopy equivalent to a one-dimensional space.

This is a short note showing how to extend a previous paper by the author and Greg Conner to the planar setting

References in corpus (1)

• The fundamental groups of subsets of closed surfaces inject into their first shape groups

Cited by in corpus (3)

• Transfinite product reduction in fundamental groupoids
• On the failure of the first Čech homotopy group to register geometrically relevant fundamental group elements
• Dense products in fundamental groupoids