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