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On quasi-small loop groups

  • Behrooz Mashayekhy , Hanieh Mirebrahimi EMAIL logo , Hamid Torabi und Ameneh Babaee
Veröffentlicht/Copyright: 9. August 2022
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Abstract

In this paper, we study some properties of homotopical closeness for paths. We define the quasi-small loop group as the subgroup of all classes of loops that are homotopically close to null-homotopic loops, denoted by π1qs(X,x) for a pointed space (X, x). Then we prove that, unlike the small loop group, the quasi-small loop group π1qs(X,x) does not depend on the base point, and that it is a normal subgroup containing π1sg(X,x), the small generated subgroup of the fundamental group. Also, we show that a space X is homotopically path Hausdorff if and only if π1qs(X,x) is trivial. Finally, as consequences, we give some relationships between the quasi-small loop group and the quasi-topological fundamental group.

  1. ( (Communicated by L'ubica Holá )

References

[1] BABAEE, A.—MASHAYEKHY, B.—MIREBRAHIMI, H.—TORABI, H.—ABDULLAHI RASHID, M.— PASHAEI, S. Z.: On topological homotopy groups and its relation to Hawaiian groups, Hacettepe J. Math. Stat. 49(4) (2020), 1437-1449.10.15672/hujms.565367Suche in Google Scholar

[2] BRAZAS, J.—FABEL, P.: On fundamental group with the quotient topology, Homotopy Relat. Struct. 10 (2015), 71–91.10.1007/s40062-013-0042-7Suche in Google Scholar

[3] BRODSKIY, N.—DYDAK, J.—LABUZ, B.—MITRA, A.: Covering maps for locally path connected spaces, Fund. Math. 218 (2012), 13–46.10.4064/fm218-1-2Suche in Google Scholar

[4] CONNER, G.—MEILSTRUP, M.—REPOVS, D.—ZASTROW, A.—ZELJKO, M.: On small homotopies of loops, Topology Appl. 155 (2008), 1089–1097.10.1016/j.topol.2008.01.009Suche in Google Scholar

[5] CANNON, J. W.—CONNER, G. R.: On the fundamental groups of one-dimensional spaces, Topology Appl. 153 (2006), 2648–2672.10.1016/j.topol.2005.10.008Suche in Google Scholar

[6] EDA, K.—KARIMOV, U. H.—REPOVS, D.: On (co)homology locally connected spaces, Topology Appl. 120(3) (2002), 397–401.10.1016/S0166-8641(01)00087-6Suche in Google Scholar

[7] EDA, K.—KARIMOV U. H.—REPOVS, D.: A construction of noncontractible simply connected cell-like two-dimensional Peano continua, Fund. Math. 195(3) (2007), 193–203.10.4064/fm195-3-1Suche in Google Scholar

[8] EDA, K.—KARIMOV U. H.—REPOVS, D.: A nonaspherical cell-like 2-dimensional simply connected continuum and related constructions, Topology Appl. 156(3) (2009), 515–521.10.1016/j.topol.2008.07.019Suche in Google Scholar

[9] EDA, K.—KARIMOV U. H.—REPOVS, D.: On the second homotopy group of SC(Z), Glas. Mat. 44 64(2) (2009), 493–498.10.3336/gm.44.2.14Suche in Google Scholar

[10] EDA, K.—KARIMOV U. H.–REPOVS, D.: On the singular homology of one class of simply-connected cell-like spaces, Mediterr. J. Math. 8(2) (2011), 153–160.10.1007/s00009-010-0079-3Suche in Google Scholar

[11] EDA, K.—KARIMOV U. H.—REPOVS, D.: On 2-dimensional nonasapherical cell-like Peano continua: A simplified approach, Mediterr. J. Math. 10(1) (2013), 519–528.10.1007/s00009-011-0165-1Suche in Google Scholar

[12] EDA, K.—KARIMOV, U. H.—REPOVS, D.—ZASTROW, A.: On Snake cones, alternating cones and related constructions, Glas. Mat. 48(1) (2013), 115–135.10.3336/gm.48.1.11Suche in Google Scholar

[13] FARBER, M.: Topology of robot motion planning. Morse theoretic methods in nonlinear analysis and in symplectic topology. NATO Sci. Ser. II Math. Phys. Chem. 217, Springer, Dordrecht, 2006, pp. 185–230.10.1007/1-4020-4266-3_05Suche in Google Scholar

[14] FISCHER, H.—REPOVS, D.—VIRK, Z.—ZASTROW, A.: On semilocally simply connected spaces, Topology Appl. 158 (2011), 397–408.10.1016/j.topol.2010.11.017Suche in Google Scholar

[15] FISCHER, H.—ZASTROW, A.: Generalized universal coverings and the shape group, Fund. Math. 197 (2007), 167–196.10.4064/fm197-0-7Suche in Google Scholar

[16] KARIMOV, U.—REPOVS, D.—ROSICKI, W.—ZASTROW, A.: On two-dimensional planar compacta not homotopically equivalent to any one-dimensional compactum, Topology Appl. 153(2–3) (2005), 284–293.10.1016/j.topol.2004.02.020Suche in Google Scholar

[17] KARIMOV U. H.—REPOVS, D.: Hawaiian groups of topological spaces, Russian Math. Surveys 61(5) (2006), 987–989.10.1070/RM2006v061n05ABEH004363Suche in Google Scholar

[18] KARIMOV U. H.—REPOVS, D.: On the homology of the Harmonic archipelago, Cent. Eur. J. Math. 10 (2012), 863–872.10.2478/s11533-012-0038-2Suche in Google Scholar

[19] MALESIC, J.—REPOVS, D.—ROSICKI, W.—ZASTROW, A.: On continua with homotopically fixed boundary, Topology Appl. 154(3) (2007), 639–654.10.1016/j.topol.2006.09.001Suche in Google Scholar

[20] MASHAYEKHY, B. H.—MIREBRAHIMI, B.—TORABI, H.—BABAEE, A. On small n-Hawaiian loops, Mediterr. J. Math. 17 (2020), 202.10.1007/s00009-020-01640-1Suche in Google Scholar

[21] PASHAEI, S. Z.—MASHAYEKHY, B.—TORABI, H.—ABDULLAHI RASHID, M.: Small Loop Transfer Spaces with respect to Subgroups of Fundamental Groups, Topology Appl. 232 (2017), 242–255.10.1016/j.topol.2017.10.012Suche in Google Scholar

[22] TORABI, H.—PAKDAMAN, A.—MASHAYEKHY, B.: Topological fundamental groups and small generated coverings, Math. Slovaca 65(5) (2015), 1153–1164.10.1515/ms-2015-0079Suche in Google Scholar

[23] VIRK, Z.: Homotopical smallness and closeness, Topology Appl. 158 (2011), 360–378.10.1016/j.topol.2010.11.010Suche in Google Scholar

[24] VIRK, Z.: Small loop spaces, Topology Appl. 157 (2010), 451–455.10.1016/j.topol.2009.10.003Suche in Google Scholar

Received: 2021-01-12
Accepted: 2021-10-05
Published Online: 2022-08-09
Published in Print: 2022-08-26

© 2022 Mathematical Institute Slovak Academy of Sciences

Heruntergeladen am 15.12.2025 von https://www.degruyterbrill.com/document/doi/10.1515/ms-2022-0071/pdf
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