Question about lower homology class of cobordism

Question

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.

Answer 1

Here is a somewhat stupid example, showing that the answer to both questions is no without some additional hypotheses.

Let $M_0 = S^1\times S^2$ and let $M_1 = S^3$. For our oriented cobordism, take $W=(S^1\times D^3)\sqcup D^4$. (It's actually a disjoint union of null-cobordisms). The generator $\alpha\in H_1(M_0;R)$ injects non-trivially into $H_1(W;R)$, and its image does not come from a class in $H_1(M_1;R)=0$.

This gives a negative answer to both questions. It's clear how to modify this to get counter-examples in any dimensions you wish. If you want a connected cobordism, just form from $W$ the connected sum of the two pieces.

Answer 2

For your first question: maybe you want to know whether the map $H_{k+1}(W,M_0\cup M_1)\to H_k(M_0)$ is surjective.

The ``answer'' is that this map sits inside the long exact sequence for the triple $(W,M_0\cup M_1,M_1)$:

$H_{k+1}(W,M_0\cup M_1)\to H_k(M_0\cup M_1,M_1)\cong H_k(M_0)\to H_k(W,M_1)$

For the second question a consideration of the long exact sequence for the pair $(W,M_0\cup M_1)$ might help:

$H_{k+1}(W,M_0\cup M_1)\to H_k(M_0\cup M_1)\to H_k(W)$

Saying that $\alpha$ and $\beta$ are the sums of generators of the homology group is in general not much information: all non-zero classes can be written in this form.