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A properly embedded topological surface is boundary-incompressible when no disk in the ambient three-manifold joins an essential arc on the topological surface to an arc on the ambient boundary, with the disk's interior disjoint from both.

Every connected closed orientable three-manifold with finite fundamental group admits spherical geometry. With cyclic fundamental group, it is a lens space.

A Heegaard diagram marks the boundaries of compressing disks from the two sides of a Heegaard surface. Attaching two-handles along the two collections, and capping resulting spherical boundary components, reconstructs the three-manifold.

A Heegaard splitting divides a closed three-manifold into two handlebodies with common boundary. For a three-manifold with boundary, one uses compression bodies, which can retain negative boundary components.

The elliptization theorem says that a closed oriented three-manifold with finite fundamental group has spherical geometry. When the fundamental group is a cyclic group, it is a lens space. Consequently the candidates are $L(5,q)$ with $q\in \{1,2,3,4\}$, using the same surgery orientation convention throughout.

The oriented classification of lens spaces says that $L(p,q)$ and $L(p,q^{\prime })$ are related by an orientation-preserving homeomorphism exactly when $q^{\prime }\equiv q^{\pm 1}(\mathrm{mod}p)$. Since $1^{-1}=1$, $4^{-1}=4$, and $2^{-1}=3$ modulo five, the complete list isThese are three distinct oriented homeomorphism classes. Reversing orientation interchanges $L(5,1)$ and $L(5,4)$; the class containing $L(5,2)$ and $L(5,3)$ admits an orientation-reversing homeomorphism. The elliptization theorem and the stated oriented classification of lens spaces are the general classification results used here.

A codimension-one foliation of a closed three-manifold is taut if every leaf meets a closed transverse curve. In particular, the fibers of a bundle over $S^1$ form a taut foliation.