Homology of a compact and orientable manifold boundary

Question

Let $X$ be a compact orientable n-manifold, $Y=\partial X$, and $R$ a ring. Suppose that $X$ is an $R$-homology ball, i.e., $H_*(X;R)\approx H_*(B^n;R)$.

The task is to compute $H_*(Y;R)$ and supposing $n=4$ and $R=\mathbb{Q}$ show that order of $H_1(Y;\mathbb{Z})$ is a square, $a^2$. Describe the number $a$ in terms of homology of $X$.

Observations:

1. Is $X$ contractible?
2. How exacty could I show $H_*(Y;R)\approx H_*(S^n;R)$ ($H_{n}(S^n;R)=R;H_{k}(S^n;R)=0,k\not=n$) using Lefschetz duality ($H_k(X,Y)=H^{n-k}(X)) $?
3. Because $ \mathbb{Q} $ is torsionfree for any space $X$: $H_*(X;\mathbb{Q})=H_*(X)\otimes \mathbb{Q} $.

As regards order of $H_1(Y;\mathbb{Z})$ and description of $a$ in terms of homology of $X$ I have no idea.

I thank you in advance for any clue.