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.