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

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.