The homology groups $$$H_{n+1} \left( \mathbb{C}\Omega_n \right)$$$

Abstract

The topic of the paper is the investigation of the homology groups of the $$$(2n+1)$$$-dimensional CW-complex $$$\mathbb{C}\Omega_n$$$. The spaces $$$\mathbb{C}\Omega_n$$$ consist of complex-valued functions and are the analogue of the spaces  $$$\Omega_n$$$, widely known in the approximation theory. The spaces $$$\mathbb{C}\Omega_n$$$ have been introduced in 2015 by A.M. Pasko who has built the CW-structure of the spaces $$$\mathbb{C}\Omega_n$$$ and using this CW-structure established that the spaces $$$\mathbb{C}\Omega_n$$$ are simply connected. Note that the mentioned CW-structure of the spaces $$$\mathbb{C}\Omega_n$$$ is the analogue of the CW-structure of the spaces $$$\Omega_n$$$ constructed by V.I. Ruban. Further A.M. Pasko found the homology groups of the space $$$\mathbb{C}\Omega_n$$$ in the dimensionalities $$$0, 1, \ldots, n, 2n-1, 2n, 2n+1$$$.  The goal of the present paper is to find the homology group $$$H_{n+1}\left ( \mathbb{C}\Omega_n \right )$$$. It is proved that $$$H_{n+1} \left ( \mathbb{C}\Omega_n \right )=\mathbb{Z}^\frac{n+1}{2}$$$ if $$$n$$$ is odd and $$$H_{n+1} \left ( \mathbb{C}\Omega_n \right )=\mathbb{Z}^\frac{n+2}{2}$$$ if $$$n$$$ is even.