Gauge theory on asymptotically periodic 4-manifolds.

*(English)*Zbl 0615.57009Since the momentous discovery by Donaldson and Freedman of the existence of an exotic differentiable structure on \({\mathbb{R}}^ 4\), progress has been made in trying to understand the space of all diffeomorphic classes on that space. Through the work of R. E. Gompf [ibid. 18, 317-328 (1983; Zbl 0496.57007); ibid. 21, 283-300 (1985; Zbl 0562.57009)] and M. H. Freedman and L. R. Taylor [ibid. 24, 69-78 (1986; Zbl 0586.57007)] more and more distinct diffeomorphism classes have been discovered. The paper under review shows that there exists an uncountable family of distinct diffeomorphism classes on \({\mathbb{R}}^ 4.\)

The idea (due apparently to Gompf) consists of taking a particular exotic \({\mathbb{R}}^ 4\), R, a homeomorphism \(\phi\) : \(R\to {\mathbb{R}}^ 4\) and considering the inverse images \(R_ r\) of balls of radius r in \({\mathbb{R}}^ 4\). Each one is homeomorphic to \({\mathbb{R}}^ 4\), but it is not clear that for \(r<s\), \(R_ r\) and \(R_ s\) are diffeomorphic. If there were such a diffeomorphism then the open submanifold \(W=\phi^{-1}(\bar B_ s\setminus B_{r-\epsilon})\) may be repeatedly attached to \(R_ s\) to give a smooth manifold which has a ”periodic end”. The analytical challenge, which the author takes up, is to provide a version of S. K. Donaldson’s argument for compact manifolds [ibid. 24, 275-341 (1986)] to these end-periodic manifolds and deduce that no such diffeomorphism can exist.

The initial problem is to define an appropriate Fredholm theory for elliptic complexes on such open manifolds and here two conditions, one involving the vanishing of an index on a compact manifold formed from W, the other a condition on the action of the first de Rham cohomology group on the complex, must be satisfied in order to define such a theory. At another stage of the argument, asymptotically end-periodic metrics are introduced (a topic which could well have a life of its own in another context). With these tools moduli spaces of self-dual connections are introduced, shown to exist and analyzed. The particular spaces used are modelled on those of R. Fintushel and R. J. Stern [ibid. 20, 523-539 (1984; Zbl 0562.53023)], and the argument pursued to its conclusion, drawing on a variety of techniques and results in analysis, gauge theory and K-theory.

The idea (due apparently to Gompf) consists of taking a particular exotic \({\mathbb{R}}^ 4\), R, a homeomorphism \(\phi\) : \(R\to {\mathbb{R}}^ 4\) and considering the inverse images \(R_ r\) of balls of radius r in \({\mathbb{R}}^ 4\). Each one is homeomorphic to \({\mathbb{R}}^ 4\), but it is not clear that for \(r<s\), \(R_ r\) and \(R_ s\) are diffeomorphic. If there were such a diffeomorphism then the open submanifold \(W=\phi^{-1}(\bar B_ s\setminus B_{r-\epsilon})\) may be repeatedly attached to \(R_ s\) to give a smooth manifold which has a ”periodic end”. The analytical challenge, which the author takes up, is to provide a version of S. K. Donaldson’s argument for compact manifolds [ibid. 24, 275-341 (1986)] to these end-periodic manifolds and deduce that no such diffeomorphism can exist.

The initial problem is to define an appropriate Fredholm theory for elliptic complexes on such open manifolds and here two conditions, one involving the vanishing of an index on a compact manifold formed from W, the other a condition on the action of the first de Rham cohomology group on the complex, must be satisfied in order to define such a theory. At another stage of the argument, asymptotically end-periodic metrics are introduced (a topic which could well have a life of its own in another context). With these tools moduli spaces of self-dual connections are introduced, shown to exist and analyzed. The particular spaces used are modelled on those of R. Fintushel and R. J. Stern [ibid. 20, 523-539 (1984; Zbl 0562.53023)], and the argument pursued to its conclusion, drawing on a variety of techniques and results in analysis, gauge theory and K-theory.

Reviewer: N.Hitchin

##### MSC:

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

53C80 | Applications of global differential geometry to the sciences |

53C05 | Connections, general theory |

58J10 | Differential complexes |

81T08 | Constructive quantum field theory |

55R10 | Fiber bundles in algebraic topology |