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.