Homomorphims from the fundamental group of planar sets

TL;DR

This paper proves that all homomorphisms from the fundamental group of planar Peano continua to certain other continua are induced by continuous maps, and constructs many non-homotopy equivalent examples with distinct fundamental groups.

Contribution

It establishes a new link between algebraic and topological properties of planar Peano continua and constructs uncountably many non-homotopy equivalent examples.

Findings

Homomorphisms are induced by continuous maps up to conjugation.

Constructs uncountably many non-homotopy equivalent planar Peano continua.

Shows these continua have non-isomorphic fundamental groups.

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.