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Embedded contact homology of prequantization bundles

Jo Nelson, Morgan Weiler

Abstract

The 2011 PhD thesis of Farris demonstrated that the ECH of a prequantization bundle over a Riemann surface is isomorphic as a Z/2Z-graded group to the exterior algebra of the homology of its base. We extend this result by computing the Z-grading on the chain complex, permitting a finer understanding of this isomorphism and a stability result for ECH. We fill in a number of technical details, including the Morse-Bott direct limit argument and the classification of certain J-holomorphic buildings. The former requires the isomorphism between filtered Seiberg-Witten Floer cohomology and filtered ECH as established by Hutchings-Taubes. The latter requires the work on higher asymptotics of pseudoholomorphic curves by Cristofaro-Gardiner--Hutchings--Zhang to obtain the writhe bounds necessary to appeal to an intersection theory argument of Hutchings-Nelson.

Embedded contact homology of prequantization bundles

Abstract

The 2011 PhD thesis of Farris demonstrated that the ECH of a prequantization bundle over a Riemann surface is isomorphic as a Z/2Z-graded group to the exterior algebra of the homology of its base. We extend this result by computing the Z-grading on the chain complex, permitting a finer understanding of this isomorphism and a stability result for ECH. We fill in a number of technical details, including the Morse-Bott direct limit argument and the classification of certain J-holomorphic buildings. The former requires the isomorphism between filtered Seiberg-Witten Floer cohomology and filtered ECH as established by Hutchings-Taubes. The latter requires the work on higher asymptotics of pseudoholomorphic curves by Cristofaro-Gardiner--Hutchings--Zhang to obtain the writhe bounds necessary to appeal to an intersection theory argument of Hutchings-Nelson.

Paper Structure

This paper contains 53 sections, 58 theorems, 294 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Definitions and overview of ECH
  3. Main theorem and outline of the proof
  4. Finer points of the isomorphism and future work
  5. Basics of ECH
  6. Pseudoholomorphic curves and currents
  7. The Fredholm index
  8. The relative first Chern number
  9. The Conley-Zehnder index in dimension 3
  10. The relative intersection pairing and relative adjunction formula
  11. Asymptotic writhe and linking number
  12. Admissible representatives
  13. The relative intersection pairing
  14. The relative adjunction formula
  15. The ECH index
  16. ...and 38 more sections

Key Result

Theorem \oldthetheorem

Let $(Y, \xi = \hbox{\em ker} \lambda)$ be a prequantization bundle over $(\Sigma_g, \omega)$ of negative Euler class $e$. Then as ${\mathbb Z}_2$-graded ${\mathbb Z}_2$-modules, Moreover, each $\Gamma\in H_1(Y;{\mathbb Z})$ satisfying $ECH_*(Y,\xi,\Gamma)\neq0$ corresponds to a number in $\{0,\dots,-e-1\}$, and under this correspondence as ${\mathbb Z}_2$-graded ${\mathbb Z}_2$-modules. When $\

Figures (1)

  • Figure 7.1: Depicted is the lattice for $e=-3$. The thicker solid lines indicate the axes while the dashed lines indicate the grid spanned by the standard lattice generated by $(1,0)$ and $(0,1)$. The $\Gamma=0$ sublattice is indicated by black points, the $\Gamma=1$ sublattice by white points, and the $\Gamma=2$ sublattice by dark grey points. The light grey triangle is $T(0,4)=T(1,3)=T(2,2)=T(3,1)=T(4,0)$.

Theorems & Definitions (123)

  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • Example \oldthetheorem
  • Remark \oldthetheorem
  • Proposition \oldthetheorem
  • Remark \oldthetheorem
  • Proposition \oldthetheorem
  • Theorem \oldthetheorem
  • Proposition \oldthetheorem
  • Proposition \oldthetheorem
  • ...and 113 more