Witten's conjecture and Property P
This important paper is aimed to prove the Property P conjecture, asserting that no homotopy 3-sphere can be obtained from \(S^3\) by Dehn surgery on a non-trivial knot. The conjecture was formulated in the early 60's by \textit{R. H. Bing} [Ann. Math. (2) 77, 210 (1963; Zbl 0115.17304)] and became one of the main open problems in knot theory (Problem 1.15 in Kirby's problem list). In the light of Perelman's recent work, it is worth observing that the property P conjecture actually follows from the Poincaré conjecture [\textit{C. McA. Gordon} and \textit{J. Luecke}, J. Am. Math. Soc. 2, No. 2, 371--415 (1989; Zbl 0678.57005)]. The authors conclude the long history of the conjecture by a masterly combination of different approaches proposed over the years together with recent results from gauge theory and symplectic topology. Denoting by \(K_r\) the 3-manifold resulting from rational \(r\)-surgery on the knot \(K\) in \(S^3\), they prove that \(\pi_1(K_1)\) admits a non-trivial representation in \(\text{SO}(3)\) for any non-trivial knot \(K\). This implies the full conjecture, the case of \(r \neq \pm1\) being already known [\textit{M. Culler}, \textit{C. McA. Gordon}, \textit{J. Luecke} and \textit{P. B. Shalen}, Ann. Math. (2) 125, 237--300 (1987; Zbl 0633.57006)]. Here is a brief outline of the argument. Assume that there exists a non-trivial knot \(K\) which contradicts the thesis. Then, the instanton Floer homology \(HF(K_1)\) is trivial and the Floer exact triangle [\textit{A. Floer}, Hofer, Helmut (ed.) et al., The Floer memorial volume. Birkhäuser Prog. Math. 133, 77--97 (1995; Zbl 0996.57515); \textit{P. J. Braam}, \textit{S. K. Donaldson}, ibid. 195--256 (1995; Zbl 0996.57516)] tells us that also \(HF(K_0)\) is trivial. On the other hand, by Gabai's proof of the Property R conjecture [\textit{D. Gabai}, J. Differ. Geom. 26, 479--536 (1987; Zbl 0639.57008)], \(K_0\) admits a taut foliation by oriented surfaces and it is not homeomorphic to \(S^1 \times S^2\). A construction of Eliashberg and Thurston [\textit{Y. M. Eliashberg} and \textit{W. P. Thurston}, Confoliations. University Lecture Series. 13. AMS (1998; Zbl 0893.53001)] allows one to convert such a foliation into a contact structure on \(K_0 \cong K_0\times\{0\}\), which is weakly compatible with a suitable symplectic form \(\omega\) defined on \(K_0\times [-1,1]\). This in turn can be extended to a closed symplectic 4-manifold \((X,\Omega)\), by a recent result of Eliashberg on concave filling [\textit{Y. Eliashberg}, Geom. Topol. 8, 277--293 (2004; Zbl 1067.53070); see also \textit{J. B. Etnyre}, Algebr. Geom. Topol. 4, 73--80 (2004; Zbl 1078.53074)]. Actually, the authors revise the Eliashberg construction to obtain some additional properties of the pair \((K_0,X)\). These properties enable them to derive from the vanishing of \(HF(K_0)\) the existence of a class \(w \in H^2(X)\) with zero Donaldson invariant \(D^w_X\) and to apply to \(X\) the weak version of Witten conjecture recently established by Feehan and Leness [\textit{P. M. N. Feehan}, \textit{T. G. Leness}, preprint, \url{arXiv:math.DG/0203047}]. Eventually the contradition arises, since the Seiberg-Witten invariants of any symplectic 4-manifold are non-trivial [\textit{C. H. Taubes}, Math. Res. Lett. 1, No. 6, 809--822 (1994; Zbl 0853.57019)] and by Witten's conjecture the same should hold for the Donaldson invariants of \(X\).
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