Homotopy type of planar continua

TL;DR

This paper demonstrates that the homotopy type of maps from Peano continua into planar or one-dimensional spaces is determined by fundamental groups, establishing new criteria for homotopy equivalence of such spaces.

Contribution

It provides a new proof that planar sets are aspherical and characterizes homotopy equivalence of planar or one-dimensional Peano continua via fundamental groups.

Findings

Homotopy type is determined by induced fundamental group homomorphism.

Planar sets are proven to be aspherical.

Homotopy equivalence corresponds to isomorphic fundamental groups.

Abstract

We prove that the homotopy type of a map from a Peano continuum into a planar or one-dimensional space is determined by the induced homomorphism of fundamental groups. This provides a new proof that planar sets are aspherical and is used to prove that two planar or one-dimensional Peano continua are homotopy equivalent if and only if they have isomorphic fundamental groups.