Inverse monoids and immersions of $\Delta$-complexes

TL;DR

This paper extends the classification of immersions between finite-dimensional connected $ riangle$-complexes by using inverse monoids, generalizing previous results from graphs and 2-dimensional complexes.

Contribution

It introduces a new approach using inverse monoids to classify immersions into $ riangle$-complexes, broadening earlier graph and 2-complex results.

Findings

Conjugacy classes of closed inverse submonoids classify connected immersions.

The method generalizes previous classifications from graphs to higher-dimensional complexes.

Provides a framework for understanding immersions via inverse monoid theory.

Abstract

An immersion $f : {\mathcal D} \rightarrow \mathcal C$ between $\Delta$-complexes is a $\Delta$-map that induces injections from star sets of $\mathcal D$ to star sets of $\mathcal C$. We study immersions between finite-dimensional connected $\Delta$-complexes by replacing the fundamental group of the base space by an appropriate inverse monoid. We show how conjugacy classes of the closed inverse submonoids of this inverse monoid may be used to classify connected immersions into the complex. This extends earlier results of Margolis and Meakin for immersions between graphs and of Meakin and Szak\'acs on immersions into $2$-dimensional $CW$-complexes.