Inverse monoids and immersions of 2-Complexes

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 [1], we extend these results to the study of immersions into 2-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 2-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. We also give a process to construct the 2-complex corresponding to a given conjugacy class given by its generators. We prove that when it is finitely generated, the process ends after a finite number of steps. This research was realized in the frames of TÁMOP 4.2.4. A/2-11-1-2012-0001 “National Excellence Program Elaborating and operating an inland student and researcher personal support system convergence program". The project was subsidized by the European Union and co-financed by the European Social Fund. This research was partially supported by the Hungarian National Foundation for Scientific Research grant no. K104251.