Inverse monoids and immersions of 2-Complexes

Abstract

Communicated by M. Sapir Dedicated to Stuart Margolis, on the occasion of his 60th birthday.It is well known that under mild conditions on a connected topological space X , connected covers of X may be classified via conjugacy classes of subgroups of the fundamental group of X . In this paper, we extend these results to the study of immersions into two-dimensional CW -complexes. An immersion f : D → C between CW -complexes is a cellular map such that each point y ∈ D has a neighborhood U that is mapped homeomorphically onto f (U ) by f . In order to classify immersions into a two-dimensional CW -complex C, we need to replace the fundamental group of C 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. C) by iteratively attaching cells of dimension n to the (n − 1)-skeleton C n−1 of C for n ≥ 1. We refer the reader to Hatcher's text [1], for the precise definition and basic properties of CW -complexes. In particular a continuous map between CWcomplexes is homotopic to a cellular map [1, Theorem 4.8], that is a continuous function that maps cells to cells of the same or lower dimension, so we will regard maps between CW -complexes as cellular maps. A subcomplex of a CW -complex is a closed subspace that is a union of cells.An immersion of a CW -complex D into a CW -complex C is a cellular map f : D → C such that each point y ∈ D has a neighborhood U which is mapped homeomorphically onto f (U ) by f . So f maps n-cells to n-cells. Thus if C is an n-dimensional CW -complex, then D is an m-dimensional CW -complex with m ≤ n. Every subcomplex of an n-dimensional CW -complex C immerses into C. Every covering space of a CW -complex C has a CW -complex structure, and every covering map is in particular an immersion.We classify connected immersions into a two-dimensional CW -complex C via conjugacy classes of closed inverse submonoids of a certain inverse monoid associated with C. The closed inverse submonoids of this inverse monoid enable us to keep track of the 1-cells and 2-cells of C that lift under the immersion, in much the same way as the subgroups of the fundamental group of C enable us to encode coverings of C. We provide an iterative process for constructing the immersion associated with a closed inverse submonoid of this inverse monoid. In many cases this iterative procedure provides an algorithm for constructing the immersion, in particular if the closed inverse submonoid is finitely generated and C has finitely many 2-cells.Section 2 of the paper outlines basic material on presentations of inverse monoids that we will need to build an inverse monoid associated with a 2-complex C. Section 3 describes an iterative procedure for constructing closed inverse submonoids of an inverse monoid from generators for the submonoid. Section 4 describes connected immersions between 2-complexes in terms of a labeling of 1-cells and 2-cells. The main results of the paper linking immersions over a ...

Graph Immersions, Inverse Monoids and Deck Transformations

“…In this section we characterize covers of graphs in terms of the concepts introduced earlier. A result related to part (a) of the following theorem was obtained by Meakin and Szakács [9] in the more general context of immersions between 2-complexes. Proof.…”

“…have been extended by Meakin and Szakács [9,10] to classify immersions between higher-dimensional cell complexes.…”

Inverse monoids and immersions of cell complexes

“…An analogous theory has been developed for graphs in [5] and for 2dimensional CW -complexes in [6]: this paper extends those results to finite dimensional ∆-complexes. However, additional care is needed to define loop monoids and prove the topological lemmas needed to establish the one-toone correspondence in the higher dimensional setting.…”

“…In his thesis [12] and paper [13], Stephen initiated the theory of presentations of inverse monoids by extending Munn's results about free inverse monoids to arbitrary presentations of inverse monoids. We refer the reader to [13] or our paper [6] for details of Stephen's construction of Schützenberger graphs and Schützenberger automata and their use in the study of presentations of inverse monoids.…”

A tribute to John Meakin on the occasion of his 75th birthday

“…In the last 15 years, John has applied inverse semigroup theory to topology and the theory of operator algebras. The papers [51,52] consider immersions over more general classes of complexes than graphs. The paper [15] is also along these lines.…”