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Inverse monoids and immersions of 2-Complexes

John Meakin

Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588, USA

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and

Nóra Szakács

Bolyai Institute, University of Szeged, Aradi Vértanúk Tere 1, H-6720 Szeged, Hungary

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https://doi.org/10.1142/S0218196715400123Cited by:5 (Source: Crossref)

Abstract

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.

Dedicated to Stuart Margolis, on the occasion of his 60th birthday.

Keywords:

AMSC: 20M18, 57M20

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