<abstract xmlns="http://www.w3.org/1999/xhtml"><p>Let <italic>G</italic> be a subgroup of the group Homeo(<italic>X</italic>) of homeomorphisms of a topological space <italic>X</italic>. Let 
<inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/awutm-2015-0015_i1.jpg"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mover accent="true"><mi>G</mi><mo>¯</mo></mover></math><tex-math>$\bar G$</tex-math></alternatives></inline-formula>
 be the closure of <italic>G</italic> in Homeo(<italic>X</italic>). The class of an orbit <italic>O</italic> of <italic>G</italic> is the union of all orbits having the same closure as <italic>O</italic>. We denote by 
<inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/awutm-2015-0015_i2.jpg"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>X</mi><mo>/</mo><mover accent="true"><mi>G</mi><mo stretchy="true">˜</mo></mover></mrow></math><tex-math>$X/\widetildeG$</tex-math></alternatives></inline-formula>
 the space of classes of orbits called the orbit class space. In this paper, we study the fundamental group of the spaces <italic>X/G</italic>, 
<inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/awutm-2015-0015_i3.jpg"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>X</mi><mo>/</mo><mover accent="true"><mi>G</mi><mo>¯</mo></mover></mrow></math><tex-math>$X/\bar G$</tex-math></alternatives></inline-formula>
 and 
<inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/awutm-2015-0015_i4.jpg"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>X</mi><mo>/</mo><mover accent="true"><mi>G</mi><mo stretchy="true">˜</mo></mover></mrow></math><tex-math>$X/\widetildeG$</tex-math></alternatives></inline-formula></p></abstract>

Let G be a subgroup of the group Homeo(X) of homeomorphisms of a topological space X. Let 
G¯$\bar G$
 be the closure of G in Homeo(X). The class of an orbit O of G is the union of all orbits having the same closure as O. We denote by 
X/G˜$X/\widetildeG$
 the space of classes of orbits called the orbit class space. In this paper, we study the fundamental group of the spaces X/G, 
X/G¯$X/\bar G$
 and 
X/G˜$X/\widetildeG$

The fundamental group of the orbit space

Laboratory of Dynamical Systems and Combinatorics, Sfax University Tunisia, Umm Alqura University, Department of Mathematics. KSA

Annals of West University of Timisoara - Mathematics and Computer Science

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"The fundamental group of the orbit space"}

Let G be a subgroup of the group Homeo(X) of homeomorphisms of a topological space X. Let 
G¯$\bar G$
 be the closure of G in Homeo(X). The class of an orbit O...

The fundamental group of the orbit space

Publié en ligne: 09 avr. 2016

Pages: 73 - 79

Reçu: 04 nov. 2015

Accepté: 22 janv. 2016

DOI: https://doi.org/10.1515/awutm-2015-0015

Mots clés
Homotopy, Fundamental group, orbit space, orbit class space

© 2015 Annals of West University of Timisoara - Mathematics

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

The fundamental group of the orbit space

Publié en ligne: 09 avr. 2016

Pages: 73 - 79

Reçu: 04 nov. 2015

Accepté: 22 janv. 2016

DOI: https://doi.org/10.1515/awutm-2015-0015

Mots clésHomotopy, Fundamental group, orbit space, orbit class space

© 2015 Annals of West University of Timisoara - Mathematics

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Mots clés
Homotopy, Fundamental group, orbit space, orbit class space