In the papers [Holomorphic disks and topological invariants for closed three-manifolds, to appear in Ann. of Math.] and [Holomorphic disks and three-manifold invariants: properties and applications, to appear in Ann. of Math.], the authors introduced a beautiful new Floer homology theory with tremendous applications; this they extended to a 4-dimensional theory in [Holomorphic triangles and invariants for smooth four-manifolds, preprint (2001)]. The current article presents a number of impressive applications of this theory in dimensions 3 and 4.
The authors begin by defining a numerical invariant for a rational homology sphere together with a Spin\(^c\) structure, which they call a correction term. This invariant is an analog of that ofK. A. Frøyshov defined in [Math. Res. Lett. 3, 373-390 (1996;Zbl 0872.57024)]. This invariant is additive with respect to connected sums and is a rational homology cobordism invariant. The authors show that for an integral homology sphere \(Y\), this correction term represents the difference between the Casson invariant of the three-manifold and the Euler characteristic of \(HF^+_{red}(Y)\).
For an integral homology 3-sphere \(Y\), the authors define thecomplexity of \(Y\) to be the rank of \(HF^+_{red}(Y)\). Previous results giving lower bounds for the complexity of a \(Y\) and for the manifold resulting from \(1/n\)-Dehn surgery on a knot in \(Y\) are improved.
An integral homology 3-sphere is said to be invisible if both the complexity and the correction term vanish; it is shown that for any knot \(K\) in an invisible homology sphere \(Y\), if the Alexander polynomial of the knot is nontrivial then any nontrivial surgery on the knot yields a 3-manifold which is not invisible. Some restrictions on knots in \(S^3\) with lens space surgeries are obtained. A table of allowed Alexander polynomials for knots in \(S^3\) with lens space surgeries is provided. Calculations are performed for the complexity, the correction term and/or \(HF^+\) for integral homology spheres resulting from surgeries on \((p,q)\)-torus knots, for certain Brieskorn spheres, for manifolds resulting from \(1/n\)-Dehn surgery on the figure eight knot in \(S^3\), for the three-torus, and for some surgies on certain pretzel knots. A new proof of Donaldson’s diagonalizability theorem is provided. If \(X\) is a four-manifold with boundary an integral homology sphere \(Y\) and if \(\xi\) is a characteristic vector for the intersection form, it is shown that \(\xi^2 + \mathrm{rk}(H^2(X;\mathbb Z))\) is bounded above by 4 times the correction term of \(Y\). This leads to a new proof of the Thom conjecture for \({\mathbb C}P^2\), first proved byP. B. Kronheimer andT. S. Mrowka in [Math Res. Lett. 1, 797-808 (1994;Zbl 0851.57023)] and byJ. W. Morgan, Z. Szabó andC. H. Taubes in [J. Differ. Geom. 44, 706-788 (1996;Zbl 0974.53063)]. The inequality also places constraints on the intersection forms of 4-manifolds which bound certain Seifert fibered spaces.

Reviewer: Cynthia L.Curtis (Ewing)

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