The homology groups of the Cartesian product $\Omega_{n_1}(m_1)\times \Omega_{n_2}(m_2)$
The paper continues the investigation of the spaces of complex-valued perfect splines $\Omega_n(m)$. These spaces were introduced as generalization of the spaces $\Omega_n$, the topology of which has been studied by V.I. Ruban, V.A. Koshcheev, A.M. Pasko. In our previous papers the homology groups o...
Saved in:
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Oles Honchar Dnipro National University
2024-12-01
|
| Series: | Researches in Mathematics |
| Subjects: | |
| Online Access: | https://vestnmath.dnu.dp.ua/index.php/rim/article/view/437/437 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1841558953516859392 |
|---|---|
| author | A.M. Pasko |
| author_facet | A.M. Pasko |
| author_sort | A.M. Pasko |
| collection | DOAJ |
| description | The paper continues the investigation of the spaces of complex-valued perfect splines $\Omega_n(m)$. These spaces were introduced as generalization of the spaces $\Omega_n$, the topology of which has been studied by V.I. Ruban, V.A. Koshcheev, A.M. Pasko. In our previous papers the homology groups of the spaces $\Omega_n(m)$ have been found and their simply connectedness was established. The topic of the paper is finding of the homology groups of the Cartesian product $\Omega_{n_1}(m_1)\times \Omega_{n_2}(m_2)$. In order to find the homology groups of this Cartesian product the Kunneth theorem has been used. Using the Kunneth theorem and the fact that $\text{Tor}(A,B)=0$ if at least one of the group $A, B$ is free we presented the homology group of the Cartesian product $\Omega_{n_1}(m_1)\times \Omega_{n_2}(m_2)$ as the sum of the tensor products of the homology groups of this spaces. Calculating the tensor products we found the homology groups of $\Omega_{n_1}(m_1)\times \Omega_{n_2}(m_2)$. |
| format | Article |
| id | doaj-art-ea0f24afac0d434585bf376e7c145470 |
| institution | Kabale University |
| issn | 2664-4991 2664-5009 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | Oles Honchar Dnipro National University |
| record_format | Article |
| series | Researches in Mathematics |
| spelling | doaj-art-ea0f24afac0d434585bf376e7c1454702025-01-05T19:38:53ZengOles Honchar Dnipro National UniversityResearches in Mathematics2664-49912664-50092024-12-0132213313610.15421/242424The homology groups of the Cartesian product $\Omega_{n_1}(m_1)\times \Omega_{n_2}(m_2)$A.M. Pasko0Oles Honchar Dnipro National UniversityThe paper continues the investigation of the spaces of complex-valued perfect splines $\Omega_n(m)$. These spaces were introduced as generalization of the spaces $\Omega_n$, the topology of which has been studied by V.I. Ruban, V.A. Koshcheev, A.M. Pasko. In our previous papers the homology groups of the spaces $\Omega_n(m)$ have been found and their simply connectedness was established. The topic of the paper is finding of the homology groups of the Cartesian product $\Omega_{n_1}(m_1)\times \Omega_{n_2}(m_2)$. In order to find the homology groups of this Cartesian product the Kunneth theorem has been used. Using the Kunneth theorem and the fact that $\text{Tor}(A,B)=0$ if at least one of the group $A, B$ is free we presented the homology group of the Cartesian product $\Omega_{n_1}(m_1)\times \Omega_{n_2}(m_2)$ as the sum of the tensor products of the homology groups of this spaces. Calculating the tensor products we found the homology groups of $\Omega_{n_1}(m_1)\times \Omega_{n_2}(m_2)$.https://vestnmath.dnu.dp.ua/index.php/rim/article/view/437/437generalized perfect splinethe cartesian producthomology groups |
| spellingShingle | A.M. Pasko The homology groups of the Cartesian product $\Omega_{n_1}(m_1)\times \Omega_{n_2}(m_2)$ Researches in Mathematics generalized perfect spline the cartesian product homology groups |
| title | The homology groups of the Cartesian product $\Omega_{n_1}(m_1)\times \Omega_{n_2}(m_2)$ |
| title_full | The homology groups of the Cartesian product $\Omega_{n_1}(m_1)\times \Omega_{n_2}(m_2)$ |
| title_fullStr | The homology groups of the Cartesian product $\Omega_{n_1}(m_1)\times \Omega_{n_2}(m_2)$ |
| title_full_unstemmed | The homology groups of the Cartesian product $\Omega_{n_1}(m_1)\times \Omega_{n_2}(m_2)$ |
| title_short | The homology groups of the Cartesian product $\Omega_{n_1}(m_1)\times \Omega_{n_2}(m_2)$ |
| title_sort | homology groups of the cartesian product omega n 1 m 1 times omega n 2 m 2 |
| topic | generalized perfect spline the cartesian product homology groups |
| url | https://vestnmath.dnu.dp.ua/index.php/rim/article/view/437/437 |
| work_keys_str_mv | AT ampasko thehomologygroupsofthecartesianproductomegan1m1timesomegan2m2 AT ampasko homologygroupsofthecartesianproductomegan1m1timesomegan2m2 |