A note on the construction of simply-connected $3$-manifolds as branched covering spaces of $S^{3}$

Abstract:

Let $K$ be a knot in ${S^3}$, and let $\omega :{\pi _1}({S^3} - K) \to {\Sigma _n}$ be a transitive representation into the symmetric group ${\Sigma _n}$ on $n$ letters. The pair $(K,\omega )$ defines a unique closed, connected orientable $3$-manifold $M(K,\omega )$, which is represented as an $n$-sheeted covering space of ${S^3}$, branched over $K$. A procedure is given for representing $M(K,\omega )$ by a Heegard splitting, and a formula is given for computing the genus of that Heegard splitting of $M(K,\omega )$. This formula is then applied to the $3$-sheeted irregular covering spaces studied by Hilden (Bull. Amer. Math. Soc. 80 (1974), 1243-1244) and Montesinos (Bull. Amer. Math. Soc. 80 (1974), 845-846), and, also, Tesis (Univ. de Madrid, 1971) to show that these particular covering spaces cannot yield counterexamples to the Poincaré Conjecture if the branch set has bridge number $< 4$.