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Smooth 4-manifolds and Symplectic Topology

In: Proceedings of the International Congress of Mathematicians

Author

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  • Robert E. Gompf

    (The University of Texas at Austin, Department of Mathematics)

Abstract

One of the famous problems of topology is the classification problem for simply connected 4-manifolds. In the context of topological manifolds (up to homeomorphism), Freedman reduced the problem in 1981 to the classification of ℤ-quadratic forms ([F]; see also [FQ]). However, for smooth manifolds (up to diffeomorphism) the problem remains wide open, and it is currently the focus of intense research. Henceforth, we only consider smooth (compact, boundaryless) manifolds. Most of our knowledge about such simply connected 4-manifolds has descended from work of Donaldson. In particular, his invariants [D] allow us for the first time to distinguish different diffeomorphism types within a given homotopy type of such manifolds — or equivalenty in this context, within a given homeomorphism type [F]. We know that many homeomorphism types each contain infinitely many diffeomorphism types (a situation that is not possible in any other dimension). This explosion of distinct examples has left topologists struggling to find order amid the confusion.

Suggested Citation

  • Robert E. Gompf, 1995. "Smooth 4-manifolds and Symplectic Topology," Springer Books, in: S. D. Chatterji (ed.), Proceedings of the International Congress of Mathematicians, pages 548-553, Springer.
    • Handle: RePEc:spr:sprchp:978-3-0348-9078-6_48
    DOI: 10.1007/978-3-0348-9078-6_48
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