The homology groups of the Cartesian product $$$\Omega_{n_1}(m_1)\times \Omega_{n_2}(m_2)$$$

Abstract

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 $$$\Omega_{n_1}(m_1)\times \Omega_{n_2}(m_2)$$$.