The fundamental group of the space $$$\Omega_n(m)$$$

A.M. Pasko (Oles Honchar Dnipro National University)

Abstract


In the present paper the spaces $$$\Omega_n(m)$$$ are considered. The spaces $$$\Omega_n(m)$$$, introduced in 2018 by A.M. Pasko and Y.O. Orekhova, are the generalization of the spaces $$$\Omega_n$$$ (the space $$$\Omega_n(2)$$$ coincides with $$$\Omega_n$$$). The investigation of homotopy properties of the spaces $$$\Omega_n$$$ has been started by V.I. Ruban in 1985 and followed by V.A. Koshcheev, A.M. Pasko. In particular V.A. Koshcheev has proved that the spaces $$$\Omega_n$$$ are simply connected. We generalized this result proving that all the spaces $$$\Omega_n(m)$$$ are simply connected. In order to prove the simply connectedness of the space $$$\Omega_n(m)$$$ we consider the 1-skeleton of this space.  Using 1-cells we form the closed ways that create the fundamental group of the space $$$\Omega_n(m)$$$. Using 2-cells we show that all these closed ways are equivalent to the trivial way. So the fundamental group of the space $$$\Omega_n(m)$$$ is trivial and the space $$$\Omega_n(m)$$$ is simply connected.

Keywords


generalized perfect spline; CW-complex; simply connected space

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References


Koshcheev V.A. "Fundamental groups of spaces of generalised perfect splines", Tr. In-ta matematiki i mekhaniki UrO RAN, 2009; 15(1): pp. 159-165. (in Russian) doi:10.1134/S0081543809060121

Pasko A.M. "The homology groups of the space $$$\Omega_n(m)$$$", Res. Math., 2019; 27(1): pp. 39-44. doi:10.15421/241904

Pasko A.M., Orekhova Y.O. "The Euler characteristic of the space $$$\Omega_n(m)$$$", Proc. Center Sci. Publ. "Veles", 5th Int. Sci. Pract. Conf. "Innov. Approaches and Modern Sci." March 2018, Kyiv; pp. 65-66.

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DOI: https://doi.org/10.15421/242207

  

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