On Boundary-Fixing Homotopies of Radal Homeomorphisms Defined on the Closed Planar Discs
1 Department of Economics, Faculty of Administration and Economics, University of Misan, Iraq.
2 Department of Mathematics, Open Educational College, Wasit, Iraq.
Research Article
Open Access Research Journal of Science and Technology, 2026, 16(01), 077-080.
Article DOI: 10.53022/oarjst.2026.16.1.0014
Publication history:
Received on 09 January 2026; revised on 14 February 2026; accepted on 17 February 2026
Abstract:
In this work, homotopies of boundary-preserving homeomorphisms defined on the planar unit disc are studied. A set of radial homeomorphisms established in polar coordinates is considered. This set is subjected to rotating the circles, which have the same center, by an angle that depends in a continuous manner on the radius. However, these maps are homotopic to the identity map, the corresponding homotopies do not inevitably fix the boundary circle. For all parameter values, a point wise-fixed homotopy is defined between the homeomorphisms and the identity map. This establishes a concrete example and develops a technique for introducing homotopies restricted to a subspace. The outcomes support exploration of boundary-preserving transformations and planar homotopies in foundations of topology.
Keywords:
Homotopy; Relative homotopy; Radial deformation; Boundary preservation
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