A <i>symplectic manifold </i>is a 2<i>n</i>-dimensional smooth manifold endowed with a closed, non-degenerate 2-form. This picks out the set of <i>Lagrangian submanifolds, n</i>-dimensional submanifolds on which the 2-form vanishes, and the group of <i>symplectomorphisms, </i>diffeomorphisms which preserve the symplectic form. In this thesis I study the homotopy type of the (compactly-supported) symplectomorphism group and the connectivity of the space of Lagrangian spheres for an array of symplectic 4-manifolds comprising some Stein surfaces and some Del Prezzo surfaces. In part I of the thesis, concerning Stein surfaces, I calculate the homotopy type of the compactly-supported symplectomorphism group for C* x C with its split symplectic form and <i>T</i>*RP<sup>2</sup> with its canonical symplectic form. More significantly, I show that the compactly-supported symplectomorphism group of the 4-dimensional <i>A<sub>n</sub></i>-Milnor fibre {<i>x</i><sup>2</sup> + <i>y</i><sup>2</sup> + <i>z<sup>n</sup></i><sup>+1</sup> = 1} is homotopy equivalent to a discrete group which injects naturally into the braid group on <i>n</i> + 1-strands. In part II of the thesis, concerning Del Pezzo surfaces: I show that the isotopy class of a Lagrangian sphere in the monotone 2-, 3- or 4-point blow-up of CP<sup>2</sup> is determined by its homology class; I calculate the homotopy type of the symplectomorphism group for the monotone 3-, 4- and 5-point blow-ups of CP<sup>2</sup>. The calculations of homotopy groups of symplectomorphism groups rely on nothing more than the standard technology of pseudoholomorphic curves and some involved topological arguments to prove the fibration property of various maps between infinite-dimensional spaces. The new idea is the compactification of the Milnor fibres by a configuration of holomorphic spheres which puts the calculation in a context familiar from the world of Lalonde-Pinsonnault and Abreu. The classification of Lagrangian spheres is based on an argument of Richard Hind.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:598911 |
| Date | January 2010 |
| Creators | Evans, J. D. |
| Publisher | University of Cambridge |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
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