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

**A.M. Pasko** (Oles Honchar Dnipro National University)

#### 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.

#### Keywords

homology group; spline; CW-complex

#### References

Koshcheev V.A. "Fundamental groups of spaces of generalised perfect splines", *Tr. In-ta matematiki i mekhaniki UrO RAN*, 2009; 15(1): pp. 159-165. (in Russian) doi:10.1134/S0081543809060121

Pasko A.M. "The homology groups of the space $$$\Omega_n(m)$$$", *Res. Math.*, 2019; 27(1): pp. 39-44. doi:10.15421/241904

Pasko A.M. "On homotopic groups of spaces of generalised perfect splines", *Res. Math.*, 2012; 17: pp. 138-140. (in Russian) doi:10.15421/241217

Pasko A.M. "Simple connectedness of one space of complex-valued functions", *Res. Math.*, 2015; 20: pp. 70-74. (in Ukrainian) doi:10.15421/241508

Pasko A.M. "The homologies of the space $$$\mathbb{C}\Omega_n$$$ for certain dimensionalities", *Res. Math.*, 2016; 21: pp. 71-76. doi:10.15421/241612

Pasko A.M., Orekhova Y.O. "The Euler characteristic of the space $$$\Omega_n(m)$$$", *Zbirnik centru naukovikh publikaciy "Veles" za materialami IV mizhnar. nauk.-prakt. konf. «Innovaciyni pidkhodi i suchasna nauka»* March, Kyiv, 2018; pp. 65-66.

Ruban V.I. "Cellular partitioning of spaces of $$$\Omega$$$-splines", *Researches on modern problems of summation and approximation of functions and their applications*, Dnipropetrovsk, 1985; pp. 39-40. (in Russian)

Ruban V.I. "Cellular structure and cohomologies of spaces of generalised perfect splines", *Res. Math.*, 1999; 4: pp. 85-90. (in Russian)

DOI:

https://doi.org/10.15421/242103
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