Assume there are three differential oriented manifold $M_0$, $M_1$, $W$ with $\partial W= M_0 \coprod -M_1$. Denote dim $M_0$=dim $M_1$=n, and dim W=n+1.

We know that for the highest homology class, we have $\partial [W]= [M_0] -[M_1]$.

My question is can we say somethings about the lower homology class?

For example, (1)For given a homology cycle $[x_0]\in H_k(M_0;R)$(For $k\leq n$), does there exist a class $[x_1]\in H_k(M_1;R)$ and a complex $Y \subset W$ such that $\partial Y= [x_0]-[x_1]$ ?

(2)Assume dim $H_k(M_0;R)=l_0$, dim $H_k(M_1;R)=l_1$.For $M_0$ take $\phi_1,\phi_2,...\phi_{l_0}$ to be the generator of $H_k(M_0;R)$, take $[\alpha]=[\phi_1]+[\phi_2]+...+[\phi_{l_0}]$. Do the same for $M_1$, take $[\beta]= [\psi_1]+[\psi_2]+...[\psi_{l_1}]$, $\psi_1,\psi_2,...,\psi_{l_1} \in H_k(M_1;R)$ are generator. Does there exist a complex $Y \subset W$ such that $\partial Y= [\alpha]-[\beta]$ ?

Thank you very much for looking at this problem.