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Title: Symplectic topology of some Stein and rational surfaces
Author: Evans, J. D.
Awarding Body: University of Cambridge
Current Institution: University of Cambridge
Date of Award: 2010
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A symplectic manifold is a 2n-dimensional smooth manifold endowed with a closed, non-degenerate 2-form. This picks out the set of Lagrangian submanifolds, n-dimensional submanifolds on which the 2-form vanishes, and the group of symplectomorphisms, 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 T*RP2 with its canonical symplectic form. More significantly, I show that the compactly-supported symplectomorphism group of the 4-dimensional An-Milnor fibre {x2 + y2 + zn+1 = 1} is homotopy equivalent to a discrete group which injects naturally into the braid group on n + 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 CP2 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 CP2. 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.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available