Embedding planar compacta in planar continua with application: homotopic maps between planar Peano continua are characterized by the fundamental group homomorphism

TL;DR

This paper characterizes when two maps between planar Peano continua are homotopic based on their induced fundamental group homomorphisms, using embedding techniques for planar compacta.

Contribution

It provides a new criterion for homotopy equivalence of maps between planar Peano continua via fundamental group homomorphisms.

Findings

Homotopic maps induce the same fundamental group homomorphism.

Planar compacta can be embedded in cellular continua with null arc sequences.

Homotopy classification reduces to algebraic fundamental group analysis.

Abstract

A planar compactum with connected complement can be an embedded in a cellular continuum by attaching a null sequence of arcs. Two based maps f and g from a planar Peano continuum X to a planar set Y are homotopic iff f and g induce the same homomorphism between fundamental groups.