Inverse monoids and immersions of Δ-complexes

2021 ◽  
pp. 1-26
Author(s):  
John Meakin ◽  
Nóra Szakács
Keyword(s):  
Fundamental Group ◽  
Conjugacy Classes ◽  
Base Space ◽  
Inverse Monoid ◽  
Inverse Monoids ◽  
Finite Dimensional

An immersion [Formula: see text] between [Formula: see text]-complexes is a [Formula: see text]-map that induces injections from star sets of [Formula: see text] to star sets of [Formula: see text]. We study immersions between finite-dimensional connected [Formula: see text]-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 [Formula: see text]-complexes.

scholarly journals Inverse monoids and immersions of 2-Complexes

2015 ◽  
Vol 25 (01n02) ◽  
pp. 301-323 ◽  
Author(s):  
John Meakin ◽  
Nóra Szakács
Keyword(s):  
Topological Space ◽  
Fundamental Group ◽  
Conjugacy Classes ◽  
Inverse Monoid ◽  
Two Dimensional ◽  
Inverse Monoids ◽  
Mild Conditions ◽  
Cw Complex ◽  
Cw 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 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.

FREE INVERSE MONOIDS AND GRAPH IMMERSIONS

1993 ◽  
Vol 03 (01) ◽  
pp. 79-99 ◽  
Author(s):  
STUART W. MARGOLIS ◽  
JOHN C. MEAKIN
Keyword(s):  
Formal Language ◽  
Inverse Monoid ◽  
Finitely Generated ◽  
Inverse Monoids ◽  
Rational Subset ◽  
The Relationship ◽  
Subgroup Theorem ◽  
Free Inverse Monoid ◽  
Natural Way

The relationship between covering spaces of graphs and subgroups of the free group leads to a rapid proof of the Nielsen-Schreier subgroup theorem. We show here that a similar relationship holds between immersions of graphs and closed inverse submonoids of free inverse monoids. We prove using these methods, that a closed inverse submonoid of a free inverse monoid is finitely generated if and only if it has finite index if and only if it is a rational subset of the free inverse monoid in the sense of formal language theory. We solve the word problem for the free inverse category over a graph Γ. We show that immersions over Γ may be classified via conjugacy classes of loop monoids of the free inverse category over Γ. In the case that Γ is a bouquet of X circles, we prove that the category of (connected) immersions over Γ is equivalent to the category of (transitive) representations of the free inverse monoid FIM(X). Such representations are coded by closed inverse submonoids of FIM(X). These monoids will be constructed in a natural way from groups acting freely on trees and they admit an idempotent pure retract onto a free inverse monoid. Applications to the classification of finitely generated subgroups of free groups via finite inverse monoids are developed.

BIRGET–RHODES EXPANSION OF GROUPS AND INVERSE MONOIDS OF MÖBIUS TYPE

2002 ◽  
Vol 12 (04) ◽  
pp. 525-533 ◽  
Author(s):  
KEUNBAE CHOI ◽  
YONGDO LIM
Keyword(s):  
Hausdorff Space ◽  
Inverse Monoid ◽  
Expansion Formula ◽  
Inverse Monoids ◽  
Isolated Point

In this paper we prove that if a group G acts faithfully on a Hausdorff space X and acts freely at a non-isolated point, then the Birget–Rhodes expansion [Formula: see text] of the group G is isomorphic to an inverse monoid of Möbius type obtained from the action.

Inverse semigroup homomorphisms via partial group actions

2001 ◽  
Vol 64 (1) ◽  
pp. 157-168 ◽  
Author(s):  
Benjamin Steinberg
Keyword(s):  
Inverse Semigroup ◽  
Group Actions ◽  
Inverse Semigroups ◽  
Inverse Monoid ◽  
Partial Group ◽  
Inverse Monoids ◽  
Maximal Group ◽  
Special Cases ◽  
Quick Proof ◽  
Group Component

This papar constructs all homomorphisms of inverse semigroups which factor through an E-unitary inverse semigroup; the construction is in terms of a semilattice component and a group component. It is shown that such homomorphisms have a unique factorisation βα with α preserving the maximal group image, β idempotent separating, and the domain I of β E-unitary; moreover, the P-representation of I is explicitly constructed. This theory, in particular, applies whenever the domain or codomain of a homomorphism is E-unitary. Stronger results are obtained for the case of F-inverse monoids.Special cases of our results include the P-theorem and the factorisation theorem for homomorphisms from E-unitary inverse semigroups (via idempotent pure followed by idempotent separating). We also deduce a criterion of McAlister–Reilly for the existence of E-unitary covers over a group, as well as a generalisation to F-inverse covers, allowing a quick proof that every inverse monoid has an F-inverse cover.

scholarly journals RELATIVE TORSION

2001 ◽  
Vol 03 (01) ◽  
pp. 15-85 ◽  
Author(s):  
DAN BURGHELEA ◽  
LEONID FRIEDLANDER ◽  
THOMAS KAPPELER
Keyword(s):  
Fundamental Group ◽  
Finite Type ◽  
Von Neumann Algebra ◽  
Hilbert Module ◽  
Von Neumann ◽  
Geometric Invariants ◽  
Finite Dimensional ◽  
Finite Von Neumann Algebra ◽  
Special Cases ◽  
Relative Torsion

This paper achieves, among other things, the following: • It frees the main result of [9] from the hypothesis of determinant class and extends this result from unitary to arbitrary representations. • It extends (and at the same times provides a new proof of) the main result of Bismut and Zhang [3] from finite dimensional representations of Γ to representations on an [Formula: see text]-Hilbert module of finite type ([Formula: see text] a finite von Neumann algebra). The result of [3] corresponds to [Formula: see text]. • It provides interesting real valued functions on the space of representations of the fundamental group Γ of a closed manifold M. These functions might be a useful source of topological and geometric invariants of M. These objectives are achieved with the help of the relative torsion ℛ, first introduced by Carey, Mathai and Mishchenko [12] in special cases. The main result of this paper calculates explicitly this relative torsion (cf. Theorem 1.1).

scholarly journals QUASI-MORPHISMS AND Lp-METRICS ON GROUPS OF VOLUME-PRESERVING DIFFEOMORPHISMS

2012 ◽  
Vol 04 (02) ◽  
pp. 255-270 ◽  
Author(s):  
MICHAEL BRANDENBURSKY
Keyword(s):  
Fundamental Group ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Volume Form ◽  
Oriented Manifold ◽  
Identity Component ◽  
Finite Dimensional ◽  
Lipschitz Embeddings ◽  
Volume Preserving

Let M be a smooth compact connected oriented manifold of dimension at least two endowed with a volume form μ. We show that every homogeneous quasi-morphism on the identity component Diff 0(M, μ) of the group of volume-preserving diffeomorphisms of M, which is induced by a quasi-morphism on the fundamental group π1(M), is Lipschitz with respect to the Lp-metric on Diff 0(M, μ). As a consequence, assuming certain conditions on π1(M), we construct bi-Lipschitz embeddings of finite dimensional vector spaces into Diff 0(M, μ).

scholarly journals Families of SU(2) representations for mapping cylinders of periodic monodromy

1997 ◽  
Vol 40 (2) ◽  
pp. 383-392
Author(s):  
G. Daskalopoulos ◽  
S. Dostoglou ◽  
R. Wentworth
Keyword(s):  
Fundamental Group ◽  
Fixed Points ◽  
Conjugacy Classes ◽  
Oriented Surface ◽  
3 Dimensional ◽  
Dimensional Mapping

We examine the action of diffeomorphisms of an oriented surface with boundary on the space of conjugacy classes of SU(2) representations of the fundamental group and prove that in the case of a single periodic diffeomorphism the induced action always has fixed points. For the corresponding 3-dimensional mapping cylinders we obtain families of representations parametrized by their value on the longitude of the torus boundary.

scholarly journals Construction of Inverse Unit Regular Monoids from a Semilattice and a Group

2018 ◽  
Vol 7 (4.36) ◽  
pp. 950
Author(s):  
Sreeja V.K
Keyword(s):  
Inverse Monoid ◽  
Inverse Monoids ◽  
Regular Semigroups ◽  
Group Of Units ◽  
Regular Monoid

This paper is a continuation of a previous paper [6] in which the structure of certain unit regular semigroups called R-strongly unit regular monoids has been studied. A monoid S is said to be unit regular if for each element s Î S there exists an element u in the group of units G of S such that s = sus. Hence where su is an idempotent and is a unit. A unit regular monoid S is said to be a unit regular inverse monoid if the set of idempotents of S form a semilattice. Since inverse monoids are R unipotent, every element of a unit regular inverse monoid can be written as s = eu where the idempotent part e is unique and u is a unit. Here we give a detailed study of inverse unit regular monoids and the results  are mainly based on [10]. The relations between the semilattice of idempotents and the group of units in unit regular inverse monoids are better identified in this case. .

scholarly journals INVOLUTORY HOPF ALGEBRAS AND 3-MANIFOLD INVARIANTS

1991 ◽  
Vol 02 (01) ◽  
pp. 41-66 ◽  
Author(s):  
GREG KUPERBERG
Keyword(s):  
Hopf Algebra ◽  
Fundamental Group ◽  
Group Algebra ◽  
Hopf Algebras ◽  
Tensor Products ◽  
State Model ◽  
Formal Expression ◽  
Finite Dimensional ◽  
Heegaard Diagram ◽  
Structure Tensors

We establish a 3-manifold invariant for each finite-dimensional, involutory Hopf algebra. If the Hopf algebra is a group algebra G, the invariant counts homomorphisms from the fundamental group of the manifold to G. The invariant can be viewed as a state model on a Heegaard diagram or a triangulation of the manifold. The computation of the invariant involves tensor products and contractions of the structure tensors of the algebra. We show that every formal expression involving these tensors corresponds to a unique 3-manifold modulo a well-understood equivalence. This raises the possibility of an algorithm which can determine whether two given 3-manifolds are homeomorphic.

Two-sided expansions of monoids

2019 ◽  
Vol 29 (08) ◽  
pp. 1467-1498 ◽  
Author(s):  
Ganna Kudryavtseva
Keyword(s):  
Inverse Monoid ◽  
Partial Action ◽  
Expansion Formula ◽  
Inverse Monoids ◽  
Universal Formula ◽  
Special Case ◽  
Free Inverse Monoid

We initiate the study of expansions of monoids in the class of two-sided restriction monoids and show that generalizations of the Birget–Rhodes prefix group expansion, despite the absence of involution, have rich structure close to that of relatively free inverse monoids. For a monoid [Formula: see text] and a class of partial actions of [Formula: see text], determined by a set, [Formula: see text], of identities, we define [Formula: see text] to be the universal [Formula: see text]-generated two-sided restriction monoid with respect to partial actions of [Formula: see text] determined by [Formula: see text]. This is an [Formula: see text]-restriction monoid which (for a certain [Formula: see text]) generalizes the Birget–Rhodes prefix expansion [Formula: see text] of a group [Formula: see text]. Our main result provides a coordinatization of [Formula: see text] via a partial action product of the idempotent semilattice [Formula: see text] of a similarly defined inverse monoid, partially acted upon by [Formula: see text]. The result by Fountain, Gomes and Gould on the structure of the free two-sided restriction monoid is recovered as a special case of our result. We show that some properties of [Formula: see text] agree well with suitable properties of [Formula: see text], such as being cancellative or embeddable into a group. We observe that if [Formula: see text] is an inverse monoid, then [Formula: see text], the free inverse monoid with respect to strong premorphisms, is isomorphic to the Lawson–Margolis–Steinberg generalized prefix expansion [Formula: see text]. This gives a presentation of [Formula: see text] and leads to a model for [Formula: see text] in terms of the known model for [Formula: see text].

Sign in / Sign up

Export Citation Format

Share Document