# An Introduction to Compactness Results in Symplectic Field by Casim Abbas

By Casim Abbas

This booklet presents an advent to symplectic box idea, a brand new and demanding topic that is presently being built. the place to begin of this concept are compactness effects for holomorphic curves tested within the final decade. the writer offers a scientific creation supplying loads of heritage fabric, a lot of that is scattered through the literature. because the content material grew out of lectures given via the writer, the most objective is to supply an access aspect into symplectic box conception for non-specialists and for graduate scholars. Extensions of sure compactness effects, that are believed to be precise by way of the experts yet haven't but been released within the literature intimately, refill the scope of this monograph.

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Table of Contents

I. normal TOPOLOGY.

1. Set concept and Logic.

2. Topological areas and non-stop Functions.

three. Connectedness and Compactness.

four. Countability and Separation Axioms.

five. The Tychonoff Theorem.

6. Metrization Theorems and Paracompactness.

7. entire Metric areas and serve as Spaces.

eight. Baire areas and size Theory.

II. ALGEBRAIC TOPOLOGY.

nine. the basic Group.

10. Separation Theorems within the Plane.

11. The Seifert-van Kampen Theorem.

12. type of Surfaces.

13. type of overlaying Spaces.

14. purposes to team Theory.

Index.

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We will combine these techniques with the results of Chapter 3 to show that, in fact, Hk (r*J is independent of g when g » k. Our starting point is once again the Riemann surface E* r of genus g with r boundary components and s punctures. £s and fix the punctures individually. 4 Computing the Cohomology of Mapping Class Groups between r* r and 39 T^'. Consider the following maps $ r > 2, * : r>2, n: r > 2, which we define as follows. For Vf, sew a pair of pants (a copy of the surface So 3 ) to the regular surface E* r along two components of its boundary.

2. According to Fenchel and Nielsen's work [1, 2], almost all hyperbolic surfaces (of finite topological type) can be obtained in this way. It is easy to see why: the parameters for the gluing are global coordinates for Teichmiiller space. To construct a surface of genus g requires 2g — 2 pairs of pants (the Euler characteristic for a pair of pants is —1). Each pair of pants has three boundaries and as such, they can be identified in pairs. There are three times | (2g — 2) gluing sites. At each site, there are two parameters, namely li and T;.

2); namely given the tangent vector fi € H (X), we can write Re(d/ c (/x)) = - / n0c. 3) 7T JX There is more to this. 4) A few comments are in order. 4). Here is how. Note that {C;} is a partition of S s giving the Fenchel-Nielsen coordinates {T,,/,}, as shown above. Thus, the twist vector fields tc{ are just the coordinate vector field g^-. The duality formula implies that u>wp is invariant under any twist flow. Putting this together with the fact that the coordinate vector fields {§7-, -gf} commute, one observes that the coefficients of LJWP in the basis {(ir,- A drj, dl{ A dr^ dl{ A dlf\ are independent of 77.