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.

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.

Homotopy Groups and CW Complexes

2020 ◽  
pp. 11-20
Author(s):  
Loring W. Tu
Keyword(s):  
Topological Space ◽  
Algebraic Topology ◽  
Weak Topology ◽  
Homotopy Theory ◽  
Fiber Bundles ◽  
Homotopy Groups ◽  
Cw Complex ◽  
Continuous Maps ◽  
Cw Complexes ◽  
Set Of Points

This chapter discusses some results about homotopy groups and CW complexes. Throughout this book, one needs to assume a certain amount of algebraic topology. A CW complex is a topological space built up from a discrete set of points by successively attaching cells one dimension at a time. The name CW complex refers to the two properties satisfied by a CW complex: closure-finiteness and weak topology. With continuous maps as morphisms, the CW complexes form a category. It turns out that this is the most appropriate category in which to do homotopy theory. The chapter also looks at fiber bundles.

The projective fundamental group of a ℤ2-shift

1995 ◽  
Vol 15 (6) ◽  
pp. 1091-1118 ◽  
Author(s):  
William Geller ◽  
James Propp
Keyword(s):  
Topological Space ◽  
Fundamental Group ◽  
Topological Conjugacy ◽  
Fundamental Groups ◽  
Two Dimensional ◽  
Limit Group ◽  
Long Distance ◽  
Mixing Properties ◽  
Point Data ◽  
Factor Maps

AbstractWe define a new invariant for symbolic ℤ2-actions, the projective fundamental group. This invariant is the limit of an inverse system of groups, each of which is the fundamental group of a space associated with the ℤ2-action. The limit group measures a kind of long-distance order that is manifested along loops in the plane, and roughly speaking bears the same relation to the mixing properties of the ℤ2-action that π1; of a topological space bears to π0. The projective fundamental group is invariant under topological conjugacy. We calculate this invariant for several important examples of ℤ2-actions, and use it to prove non-existence of certain constant-to-one factor maps between two-dimensional subshifts. Subshifts that have the same entropy and periodic point data can have different projective fundamental groups.

scholarly journals Spaces which Invert Weak Homotopy Equivalences

2018 ◽  
Vol 62 (2) ◽  
pp. 553-558
Author(s):  
Jonathan Ariel Barmak
Keyword(s):  
Empty Space ◽  
Weak Equivalence ◽  
Homotopy Equivalence ◽  
Homotopy Classes ◽  
Cw Complex ◽  
Cw Complexes ◽  
Weak Homotopy ◽  
Weak Homotopy Equivalence

AbstractIt is well known that if X is a CW-complex, then for every weak homotopy equivalence f : A → B, the map f* : [X, A] → [X, B] induced in homotopy classes is a bijection. In fact, up to homotopy equivalence, only CW-complexes have that property. Now, for which spaces X is f* : [B, X] → [A, X] a bijection for every weak equivalence f? This question was considered by J. Strom and T. Goodwillie. In this note we prove that a non-empty space inverts weak equivalences if and only if it is contractible.

scholarly journals A Lemma on Continuous Functions

1960 ◽  
Vol 3 (2) ◽  
pp. 186-187
Author(s):  
J. Lipman
Keyword(s):  
Topological Space ◽  
Fundamental Group ◽  
Closed Interval ◽  
Continuous Functions ◽  
The Other ◽  
Special Consideration ◽  
Boundary Lines

The point of this note is to get a lemma which is useful in treating homotopy between paths in a topological space [1].As explained in the reference, two paths joining a given pair of points in a space E are homotopic if there exists a mapping F: I x I →E (I being the closed interval [0,1] ) which deforms one path continuously into the other. In practice, when two paths are homotopic and the mapping F is constructed, then the verification of all its required properties, with the possible exception of continuity, is trivial. The snag occurs when F is a combination of two or three functions on different subsets of I x I. Then the boundary lines between these subsets have to be given special consideration, and although the problems resulting are routine their disposal can involve some tedious calculation and repetition. In the development [l] of the fundamental group of a space, for example, this sort of situation comes up four or five times.

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.

Facile Synthesis of 4,5-Dihydro-1,3,4-Thiadiazoles by 1,3-Dipolar Cycloaddition of Thioisomünchnones

2009 ◽  
Vol 62 (4) ◽  
pp. 356 ◽  
Author(s):  
Bárbara Sánchez ◽  
José Luis Bravo ◽  
María Josí Arívalo ◽  
Ignacio López ◽  
Mark E. Light ◽  
...  
Keyword(s):  
Structure Analysis ◽  
Structure Elucidation ◽  
Room Temperature ◽  
Dipolar Cycloaddition ◽  
Two Dimensional ◽  
Facile Synthesis ◽  
X Ray ◽  
Mild Conditions ◽  
Nmr Techniques

The present paper summarizes a straightforward synthesis of 4,5-dihydro-1,3,4-thiadiazoles by the 1,3-dipolar cycloaddition of thioisomünchnones. These reactions have been carried out in dichloromethane and are essentially complete within 60 min at room temperature. Under such mild conditions the asymmetric version has been explored as well. Unequivocal structure elucidation has been accomplished by means of one- and two-dimensional NMR techniques as well as X-ray structure analysis.

Finiteness conditions for CW complexes. Il

1966 ◽  
Vol 295 (1441) ◽  
pp. 129-139 ◽  
Keyword(s):  
Topological Space ◽  
Finite Number ◽  
Simple Form ◽  
Finite Dimension ◽  
Homotopy Equivalent ◽  
Finite Complex ◽  
Finiteness Conditions ◽  
Cw Complexes ◽  
Number Of Cells

A CW complex is a topological space which is built up in an inductive way by a process of attaching cells. Spaces homotopy equivalent to CW complexes play a fundamental role in topology. In the previous paper with the same title we gave criteria (in terms of more-or-less standard invariants of the space) for a CW complex to be homotopy equivalent to one of finite dimension, or to one with a finite number of cells in each dimension, or to a finite complex. This paper contains some simplification of these results. In addition, algebraic machinery is developed which provides a rough classification of CW complexes homotopy equivalent to a given one (the existence clause of the classification is the interesting one). The results would take a particularly simple form if a certain (rather implausible) conjecture could be established.

The forcing partial order on the three times punctured disk

1997 ◽  
Vol 17 (3) ◽  
pp. 593-610 ◽  
Author(s):  
MICHAEL HANDEL
Keyword(s):  
Partial Order ◽  
Mapping Class Group ◽  
Class Group ◽  
Conjugacy Classes ◽  
Explicit Description ◽  
Mapping Class ◽  
Two Dimensional

The two-dimensional analogue of the Sharkovski order on periods for maps of the interval restricts to a partial order on essential pseudo-Anosov conjugacy classes in the mapping class group of the $n$-times punctured disk. In this paper we give an explicit description of this restricted partial order in the case when $n=3$.

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