From symplectic deformation to isotopy. (English) Zbl 0928.57018

Stern, Ronald J. (ed.), Topics in symplectic \(4\)-manifolds. 1st International Press lectures presented in Irvine, CA, USA, March 28–30, 1996. Cambridge, MA: International Press. First Int. Press Lect. Ser. 1, 85-99 (1998).
Two symplectic forms on \(X\) are said to be deformation equivalent if they may be joined by a family of symplectic forms, and are called isotopic if this family may be chosen so that its elements all lie in the same cohomology class. There are no examples of cohomologous symplectic forms that are deformation equivalent but not isotopic in dimension 4, and in this note the author studies a possibility that the two notions are the same in this case. In [F. Lalonde, Math. Ann. 300, No. 2, 273-296 (1994; Zbl 0812.53032)] Lalonde and the author found an “inflation” procedure that converts a deformation into an isotopy, and they applied it to establish the uniqueness of symplectic structure on ruled surfaces. The present note extends the range of this procedure and describes various applications of it. Let \(X\) be an oriented 4-manifold which does not have simple SW-type. The author shows that any two cohomologous and deformation equivalent symplectic forms on \(X\) are isotopic.
For the entire collection see [Zbl 0906.00020].


57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
57R57 Applications of global analysis to structures on manifolds
57R55 Differentiable structures in differential topology
53D99 Symplectic geometry, contact geometry


Zbl 0812.53032
Full Text: arXiv