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**Polynomial invariants for smooth four-manifolds.**
*(English)*
Zbl 0715.57007

The classical invariants for smooth manifolds which have been so successful for the classification of manifolds in higher dimensions have not been effective for the classification of smooth 4-manifolds. In the early 1980’s the author introduced to the study of 4-manifolds dramatically new methods using Yang-Mills and gauge theory to obtain restrictions on the intersection forms of smooth 4-manifolds [J. Differ. Geom. 18, 279–315 (1983; Zbl 0507.57010)]. These restrictions are not detected by classical invariants. The Yang-Mills equations depend on the Riemannian geometry of the underlying 4-manifold; however, certain homological properties of the moduli space of solutions are invariant under continuous change of metric. Since any two metrics can be joined by a path, these homological properties depend only upon the underlying smooth manifold and furnish potentially new differential topological invariants. This point of view was first developed in earlier pioneering work of the author [ibid. 26, 141–168 (1987; Zbl 0631.57010)]. Invariants were defined for manifolds with \(b^ 2_+\)- the rank of a maximal positive subspace for the intersection form - equal to 1. These invariants were shown to distinguish distinct smooth structures on homeomorphic 4-manifolds. Later, R. Friedman and J. W. Morgan [ibid. 27, 297–398 (1988; Zbl 0669.57016 and Zbl 0669.57017)] and C. Okonek and A. Van de Ven [Invent. Math. 86, 357–370 (1986; Zbl 0613.14018)] showed that these invariants can distinguish mutually distinct differentiable structures on an infinite family of homeomorphic 4-manifolds.

The paper under review extends these ideas. For simply-connected smooth 4-manifolds \(X\) with \(b^ 2_+>1\) and odd, the author defines an infinite set of differential topological invariants (which have become known as Donaldson invariants) for \(X\) which are distinguished elements \(q_{k,X}\), \(k\in {\mathbb{Z}}\), of the ring \(S^*(H_ 2(X))\) of polynomials in the homology of \(X\). He then uses these invariants to obtain spectacular results concerning the differential topology of algebraic surfaces. The most striking results are:

Theorem A. Let \(S\) be a simply-connected, smooth, complex projective surface. If \(S\) is orientation preserving diffeomorphic to the connected sum of two oriented 4-manifolds, then one of these manifolds has a negative definite intersection form.

This follows immediately from two other theorems: Theorem B. Suppose \(X\) is a simply-connected, oriented smooth 4-manifold with \(b^ 2_+\) odd and \(X\) is orientation preserving diffeomorphic to the connected sum of two oriented 4-manifolds where the \(b^ 2_+\) of both manifolds is strictly positive. Then all the Donaldson invariants for \(X\) vanish.

Theorem C. Let \(S\) be a simply-connected complex projective surface and \(H\) a hyperplane class in \(H_ 2(S)\). Then for a large enough \(k\), \(q_{k,S}(H,\dots,H)>0.\)

These results are impressive, as are their proofs. The proofs are an artful and demanding blend of differential and algebraic topology and geometry and local and global analysis. This paper is a mine for technique and ideas and, together with the author’s paper in J. Differ. Geom. 24, 275–341 (1986; Zbl 0635.57007) and the new book by P. Kronheimer and the author [The geometry of four-manifolds, Oxford: Clarendon Press (1990; Zbl 0820.57002)], make for indispensible resources for the modern study of smooth 4-manifolds.

The paper under review extends these ideas. For simply-connected smooth 4-manifolds \(X\) with \(b^ 2_+>1\) and odd, the author defines an infinite set of differential topological invariants (which have become known as Donaldson invariants) for \(X\) which are distinguished elements \(q_{k,X}\), \(k\in {\mathbb{Z}}\), of the ring \(S^*(H_ 2(X))\) of polynomials in the homology of \(X\). He then uses these invariants to obtain spectacular results concerning the differential topology of algebraic surfaces. The most striking results are:

Theorem A. Let \(S\) be a simply-connected, smooth, complex projective surface. If \(S\) is orientation preserving diffeomorphic to the connected sum of two oriented 4-manifolds, then one of these manifolds has a negative definite intersection form.

This follows immediately from two other theorems: Theorem B. Suppose \(X\) is a simply-connected, oriented smooth 4-manifold with \(b^ 2_+\) odd and \(X\) is orientation preserving diffeomorphic to the connected sum of two oriented 4-manifolds where the \(b^ 2_+\) of both manifolds is strictly positive. Then all the Donaldson invariants for \(X\) vanish.

Theorem C. Let \(S\) be a simply-connected complex projective surface and \(H\) a hyperplane class in \(H_ 2(S)\). Then for a large enough \(k\), \(q_{k,S}(H,\dots,H)>0.\)

These results are impressive, as are their proofs. The proofs are an artful and demanding blend of differential and algebraic topology and geometry and local and global analysis. This paper is a mine for technique and ideas and, together with the author’s paper in J. Differ. Geom. 24, 275–341 (1986; Zbl 0635.57007) and the new book by P. Kronheimer and the author [The geometry of four-manifolds, Oxford: Clarendon Press (1990; Zbl 0820.57002)], make for indispensible resources for the modern study of smooth 4-manifolds.

Reviewer: Ronald J. Stern

### MSC:

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

32J15 | Compact complex surfaces |

57R55 | Differentiable structures in differential topology |

57R50 | Differential topological aspects of diffeomorphisms |

57R80 | \(h\)- and \(s\)-cobordism |

58J20 | Index theory and related fixed-point theorems on manifolds |

58J40 | Pseudodifferential and Fourier integral operators on manifolds |