Lickorish–Wallace theorem

Remarks

Wallace and Lickorish proved the theorem independently, by different methods. Idea of Lickorish's proof: write $M$ as a Heegaard splitting whose gluing map differs from that of a standard splitting of $S^3$ by a homeomorphism of the Heegaard surface. By the Dehn–Lickorish theorem, this homeomorphism is isotopic to a product of Dehn twists. Each Dehn twist can be realised by a $\pm 1$ surgery on a copy of its curve pushed slightly off the surface into $S^3$, and such a copy is unknotted.

The theorem at work: $S^2 \times S^1$ is $0$-surgery on the unknot, and the lens space $L(p, 1)$ is integral surgery with coefficient $\pm p$ on the unknot, depending on the orientation convention. The theorem implies that every closed orientable $3$-manifold bounds a compact orientable $4$-manifold, obtained from the $4$-ball by attaching $2$-handles along $L$.

Going deeper

For a student

The theorem is the starting point of the description of $3$-manifolds by framed links, and of Kirby calculus, which says when two framed links give the same manifold. Quantum invariants of $3$-manifolds, such as the Witten–Reshetikhin–Turaev invariants, are defined from link invariants through such surgery presentations. Its $4$-dimensional reading, that every $3$-manifold bounds a $4$-manifold built with $2$-handles, connects it with the vanishing of the oriented cobordism group in dimension $3$.