##
**Donaldson invariants for connected sums along surfaces of genus 2.**
*(English)*
Zbl 0957.57023

In [Topology 29, No. 3, 257-315 (1990; Zbl 0715.57007)], S. K. Donaldson defined his famous smooth invariants on 4-manifolds through gauge theory. He proved two fundamental results in that paper: (1) a vanishing result for 4-manifolds splitting along \(S^3\), and (2) a nonvanishing result for algebraic surfaces. For a while, understanding the Donaldson invariants was a difficult task since their definition is involved with complicated constructions among topology and geometry of (anti-)self-dual connection spaces and nonlinear analysis.

On the 1990 ICM, P. B. Kronheimer proposed a new program to study the Donaldson invariants via embedded surfaces (without bothering a few coefficients from extremely hard algebraic geometry calculations) [Proc. Int. Congr. Math. Kyoto/Japan 1990, Vol. I, 529-539 (1991; Zbl 0746.53041)]. Later, in a sequence of collaborating work, Kronheimer and Mrowka studied the gauge theory for embedded surfaces through singular (anti-)self-dual connection spaces and the Mayer-Vietoris principle under glueing processes. Two important consequences are the proof of the Milnor unknotting number conjecture and the generalized Thom conjecture for some smooth 4-manifolds and smooth embedded surfaces other than spheres with self-intersection \(-1\) or inessential spheres with self-intersection \(0\). These lead them to understand the general structure of the Donaldson invariants and the important notions of simple type and basic class [P. B. Kronheimer and T. S. Mrowka, J. Differ. Geom. 41, No. 3, 573-734 (1995; Zbl 0842.57022); R. Fintushel and R. J. Stern, ibid. 42, No. 3, 577-633 (1995; Zbl 0863.57021)].

Among other things, we would like to mention two consequences. (1) The study of the Donaldson \(SU(2)\)-invariants reflects the study of the Seiberg-Witten \(U(1)\)-monopole theory as a duality from high energy physics point of view, the Seiberg-Witten (abelian) theory is much easier in many cases. In particular, Kronheimer and Mrowka used the Seiberg-Witten \(U(1)\)-monopole to complete the proof of the “Thom conjecture” about the genus of smooth algebraic curves in \(CP^2\). (2) Using the embedded spheres of self-intersection \(-2\) and \(-3\), R. Fintushel and R. J. Stern gave a beautiful blowup formula for the Donaldson invariants in terms of elliptic functions [Ann. Math. (2) 143, No. 3, 529-546 (1996; Zbl 0869.57019)] as well as rational blowdown relations of the Donaldson invariants in the presence of embedded spheres [J. Differ. Geom. 46, No. 2, 181-235 (1997; Zbl 0896.57022)].

The work of Morgan, Mrowka and Ruberman was to study the vanishing property of the Donaldson invariants with embedded torus (genus \(1\) case) of self-intersection \(\geq 2\). J. W. Morgan and Z. Szabó studied embedded surfaces with genus \(2\) in 4-manifolds and the behavior of the Donaldson invariants [Duke Math. J. 89, No. 3, 577-602 (199 Zbl 0886.57016)]. The paper under review presents a general result for the embedded genus \(2\) surfaces which grows out from the author’s thesis. The study of this type of problems is to split admissible 4-manifolds along 3-manifold \(Y = \Sigma \times S^1\), where \(\Sigma\) is the embedded surface (genus \(2\) in this case), and to analyze the Mayer-Vietoris principle for the Donaldson invariants. The general framework is to use the relative Donaldson-Floer theory to obtain nontrivial contributions from Floer-cycles (obstruction bundles on the overlaps). The truncated Floer (co)homology plays an essential role for the circle bundles over surfaces. In order to obtain the Donaldson invariants for crossing homology classes on the neck, the author, suggested by Mrowka, used the Fukaya-Floer homology to capture the full information after the glueing process.

First of all, the author uses examples as the \(K3\) surface and \(\mathbb{C} P^1\times\Sigma\) to show that there are four linearly independent functions over a certain ring that appeared in the glueing formula for the Donaldson invariants, then spanning the Donaldson invariants under simple type condition gives the constraints for the coefficients of those four functions (basis) (one of them vanishes, the others are from contributions of basic classes intersect the surface \(\Sigma\) of \(-2, 0, 2\)). The expression of the glueing formula is universal so that one can use some known examples to compute the coefficients. This leads to the proof of the main theorem 3 in the paper.

On the 1990 ICM, P. B. Kronheimer proposed a new program to study the Donaldson invariants via embedded surfaces (without bothering a few coefficients from extremely hard algebraic geometry calculations) [Proc. Int. Congr. Math. Kyoto/Japan 1990, Vol. I, 529-539 (1991; Zbl 0746.53041)]. Later, in a sequence of collaborating work, Kronheimer and Mrowka studied the gauge theory for embedded surfaces through singular (anti-)self-dual connection spaces and the Mayer-Vietoris principle under glueing processes. Two important consequences are the proof of the Milnor unknotting number conjecture and the generalized Thom conjecture for some smooth 4-manifolds and smooth embedded surfaces other than spheres with self-intersection \(-1\) or inessential spheres with self-intersection \(0\). These lead them to understand the general structure of the Donaldson invariants and the important notions of simple type and basic class [P. B. Kronheimer and T. S. Mrowka, J. Differ. Geom. 41, No. 3, 573-734 (1995; Zbl 0842.57022); R. Fintushel and R. J. Stern, ibid. 42, No. 3, 577-633 (1995; Zbl 0863.57021)].

Among other things, we would like to mention two consequences. (1) The study of the Donaldson \(SU(2)\)-invariants reflects the study of the Seiberg-Witten \(U(1)\)-monopole theory as a duality from high energy physics point of view, the Seiberg-Witten (abelian) theory is much easier in many cases. In particular, Kronheimer and Mrowka used the Seiberg-Witten \(U(1)\)-monopole to complete the proof of the “Thom conjecture” about the genus of smooth algebraic curves in \(CP^2\). (2) Using the embedded spheres of self-intersection \(-2\) and \(-3\), R. Fintushel and R. J. Stern gave a beautiful blowup formula for the Donaldson invariants in terms of elliptic functions [Ann. Math. (2) 143, No. 3, 529-546 (1996; Zbl 0869.57019)] as well as rational blowdown relations of the Donaldson invariants in the presence of embedded spheres [J. Differ. Geom. 46, No. 2, 181-235 (1997; Zbl 0896.57022)].

The work of Morgan, Mrowka and Ruberman was to study the vanishing property of the Donaldson invariants with embedded torus (genus \(1\) case) of self-intersection \(\geq 2\). J. W. Morgan and Z. Szabó studied embedded surfaces with genus \(2\) in 4-manifolds and the behavior of the Donaldson invariants [Duke Math. J. 89, No. 3, 577-602 (199 Zbl 0886.57016)]. The paper under review presents a general result for the embedded genus \(2\) surfaces which grows out from the author’s thesis. The study of this type of problems is to split admissible 4-manifolds along 3-manifold \(Y = \Sigma \times S^1\), where \(\Sigma\) is the embedded surface (genus \(2\) in this case), and to analyze the Mayer-Vietoris principle for the Donaldson invariants. The general framework is to use the relative Donaldson-Floer theory to obtain nontrivial contributions from Floer-cycles (obstruction bundles on the overlaps). The truncated Floer (co)homology plays an essential role for the circle bundles over surfaces. In order to obtain the Donaldson invariants for crossing homology classes on the neck, the author, suggested by Mrowka, used the Fukaya-Floer homology to capture the full information after the glueing process.

First of all, the author uses examples as the \(K3\) surface and \(\mathbb{C} P^1\times\Sigma\) to show that there are four linearly independent functions over a certain ring that appeared in the glueing formula for the Donaldson invariants, then spanning the Donaldson invariants under simple type condition gives the constraints for the coefficients of those four functions (basis) (one of them vanishes, the others are from contributions of basic classes intersect the surface \(\Sigma\) of \(-2, 0, 2\)). The expression of the glueing formula is universal so that one can use some known examples to compute the coefficients. This leads to the proof of the main theorem 3 in the paper.

Reviewer: Weiping Li (Stillwater)

### MSC:

57R57 | Applications of global analysis to structures on manifolds |

57N13 | Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010) |