# Subdivision of Maps of Digital Images

## TL;DR

This paper investigates subdivision techniques for digital images to develop invariants like the digital fundamental group that are less rigid and more aligned with classical topological homotopy invariance.

## Contribution

It establishes foundational results on subdividing digital maps, enabling the digital fundamental group to be an invariant under a less rigid equivalence relation.

## Key findings

- Fundamental results on subdivision of digital maps with 1- or 2-dimensional domains.
- Digital fundamental group can be invariant under a less rigid equivalence.
- Lays groundwork for defining other invariants with topological homotopy invariance.

## Abstract

With a view towards providing tools for analyzing and understanding digitized images, various notions from algebraic topology have been introduced into the setting of digital topology. In the ordinary topological setting, invariants such as the fundamental group are invariants of homotopy type. In the digital setting, however, the usual notion of homotopy leads to a very rigid invariance that does not correspond well with the topological notion of homotopy invariance. In this paper, we establish fundamental results about subdivision of maps of digital images with $1$- or $2$-dimensional domains. Our results lay the groundwork for showing that the digital fundamental group is an invariant of a much less rigid equivalence relation on digital images, that is more akin to the topological notion of homotopy invariance. Our results also lay the groundwork for defining other invariants of digital images in a way that makes them invariants of this less rigid equivalence.