Consider a generator $x$ of the singular homology group $H_3(S^3)$. I think of this (perhaps wrongly?) as something like the identity on $S^3$, cut up into simplices. Now we have the Hopf fibration $\eta: S^3 \to S^2$, which gives us $\eta_*x \in H_3(S^2) = 0$. Thus $\eta_*x$ is a boundary. Of what is it the boundary?
If this is silly or uninteresting, why is it so?
$\newcommand{\Cpx}{\mathbf{C}}\newcommand{\Proj}{\mathbf{P}}$Let $M$ be the complex surface obtained by blowing up the unit ball in $\Cpx^{2}$ at the origin. The exceptional curve at the origin is a complex projective line, i.e., an $S^{2}$. The boundary of $M$ is $S^{3}$, and radial projection from the unit ball to the origin induces a holomorphic (hence continuous) map $M \to S^{2}$ whose restriction to the boundary is the Hopf map $S^{3} \to S^{2}$. Thus $\eta_{*}x$ may be viewed as the boundary of $M$.
