Embedded homology sphere

Question

Let $Y$ be a rational homology three sphere, and $Y\hookrightarrow S^1\times S^3$ a smooth embedding.

Q. Can we say that $[Y]$ is a generator of $H_3(S^1\times S^3;\mathbb Z)$?

Answer 1

Suppose $Y$ is the standard three sphere. Since $S^1\times S^3$ is a four dimensional manifold, it contains an embedded $\mathbb{R}^4$ (a chart). Just embed $Y$ into this $\mathbb{R}^4$.

Answer 2

If it's nonseparating, then it is a generator. This is a straightforward consequence of Poincare duality. If it separates then of course it is 0 in homology.