View $S^3$ as a principal circle bundle over $S^2$ associated to a degree one complex line bundle. Using real oriented blowups at a point $s \in S^2$, we may construct a topological disc $D^2 \subset S^3$, whose boundary is the circle fiber $F_s$ over $s$, which away from the boundary defines a section of the circle bundle over $S^2 \setminus s$.
Consider the bundle compatible composition $S^3 \to L(1,p)$, given by quotienting by $\mu_p$ in the fibers. This gives rise to an immersion of the disc $f: D^2 \to L(1,p)$. The image of $f$, $im(f)$ is also the closure of a section of the quotiented circle bundle defined away from $s$. How does one describe $im(f)$ as a topological space. In my head, this is a topological disc but that is impossible since that seems make the fiber of $L(1,p) \to S^2$ a boundary, which is impossible for homological reasons.