# Embedded homology sphere

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 $H_3(S^1\times S^3;\mathbb Z)$?

• Of course not. Perhaps you mean to ask a more refined question? – Ryan Budney Dec 17 '17 at 18:27

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 $\mathbb{R}^4$.