Inverse monoids and immersions of cell complexes

Abstract

An immersion f : D → C between cell complexes is a local homeomorphism onto its image that commutes with the characteristic maps of the cell complexes. We study immersions between finitedimensional connected ∆-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 Szakács on immersions into 2-dimensional CW -complexes.