On the homology groups $H_k(\mathbb{C}\Omega_n)$, $k=1, ..., n$

Abstract

In the paper the homology groups of the $(2n+1)$-dimensional CW-complex $\mathbb{C}\Omega_n$ are investigated. The spaces $\mathbb{C}\Omega_n$ consist of complex-valued functions and generalize the widely known in the approximation theory spaces $\Omega_n$. The research of the homotopy properties of the spaces $\Omega_n$ has been started by V.I. Ruban who in 1985 found the n-dimensional homology group of the space $\Omega_n$ and in 1999 found all the cohomology groups of this space. The spaces $\mathbb{C}\Omega_n$ have been introduced by A.M. Pasko who in 2015 has built the structure of CW-complex on these spaces. This CW-structure is analogue of the CW-structure of the space $\Omega_n$ introduced by V.I. Ruban. In present paper in order to investigate the homology groups of the spaces $\mathbb{C}\Omega_n$ we calculate the relative homology groups $H_k(\mathbb{C}\Omega_n, \mathbb{C}\Omega_{n-1})$, it turned out that the groups $H_k \left (\mathbb{C}\Omega_n, \mathbb{C}\Omega_{n-1} \right )$ are trivial if $1\leq k < n$ and $H_k \left (\mathbb{C}\Omega_n, \mathbb{C}\Omega_{n-1} \right )=\mathbb{Z}^{C^{k-n}_{n+1}}$ if $n \leq k \leq 2n+1$, in particular $H_n \left (\mathbb{C}\Omega_n, \mathbb{C}\Omega_{n-1} \right )=\mathbb{Z}$. Further we consider the exact homology sequence of the pair $\left (\mathbb{C}\Omega_{n+1}, \mathbb{C}\Omega_n \right )$ and prove that its inclusion operator $i_*: H_k(\mathbb{C}\Omega_n) \rightarrow H_k(\mathbb{C}\Omega_{n+1})$ is zero. Taking into account that the relative homology groups $H_k \left (\mathbb{C}\Omega_{n+1}, \mathbb{C}\Omega_n \right )$ are zero if $1\leq k \leq n$ and the inclusion operator $i_*=0$ we have derived from the exact homology sequence of the pair $\left (\mathbb{C}\Omega_{n+1}, \mathbb{C}\Omega_n \right )$ that the homology groups $H_k \left ( \mathbb{C}\Omega_n \right ), 1\leq k<n,$ are trivial. The similar considerations made it possible to calculate the group $H_n(\mathbb{C}\Omega_n)$. So the homology groups $H_k(\mathbb{C}\Omega_n), n \geq 2, k=1,...,n,$ have been found.