Property P conjecture

Remarks

The homology of $S^3_{p/q}(K)$ is $\Z / p$, or $\Z$ when $p = 0$, so it is nontrivial unless $p = \pm 1$. The content of the theorem is therefore about the slopes $r = \pm 1/q$. Kronheimer and Mrowka proved it in 2004 using instanton Floer homology and work on taut foliations and contact structures. With the Poincaré conjecture, proved by Perelman, it also follows from the Gordon–Luecke result that a nontrivial surgery on a nontrivial knot never gives $S^3$. The hypothesis that $K$ is knotted is needed: $1/q$ surgery on the unknot gives $S^3$. Example: $+1$ surgery on the right-handed trefoil gives the Poincaré homology sphere, up to orientation, which has the same homology as $S^3$ but a fundamental group of order $120$.

Going deeper

For a student

By the Lickorish–Wallace theorem every closed orientable $3$-manifold arises by surgery on a link, so it is natural to ask what surgery on a single knot can produce. Property P says that, apart from the slope $\infty$ which gives back $S^3$, it never produces a fake or real $3$-sphere. It implies the key step of the Gordon–Luecke theorem, that nontrivial surgery on a nontrivial knot never gives $S^3$. Conversely, that step together with the Poincaré conjecture implies Property P. The proof by Kronheimer and Mrowka shows that the fundamental group of $S^3_{\pm 1/q}(K)$ has a nontrivial representation into $SU(2)$.