[Paper Review] Embedded contact homology of prequantization bundles

J. M. Nelson, Morgan Weiler|arXiv (Cornell University)|Jul 27, 2020

Geometric and Algebraic Topology33 references4 citations

TL;DR

This paper computes the embedded contact homology (ECH) of prequantization bundles over closed Riemann surfaces by refining the Z-grading on the ECH chain complex, extending Farris's 2011 result. It establishes a Z-graded isomorphism between the ECH of such bundles and the exterior algebra of the base's homology, using filtered ECH, direct limits, and a detailed classification of J-holomorphic buildings via writhe bounds and intersection theory.

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.

Motivation & Objective

• To refine the Z-grading on the embedded contact homology (ECH) chain complex of prequantization bundles over Riemann surfaces, going beyond the prior Z2-grading result.
• To provide a complete and rigorous proof of the isomorphism between ECH and the exterior algebra of the base’s homology, including the grading structure.
• To fill technical gaps in the Morse-Bott direct limit argument and classify J-holomorphic buildings in the context of prequantization bundles.
• To establish a stability result for ECH by controlling contributions from higher-genus curves using writhe bounds and intersection theory.
• To lay the groundwork for future computation of U-maps and spectral invariants in Seifert fiber spaces and prequantization bundles.

Proposed method

• Uses filtered ECH and direct limits, relying on the isomorphism between filtered Seiberg-Witten Floer cohomology and filtered ECH established by Hutchings-Taubes (2013).
• Applies domain-dependent almost complex structures J to control holomorphic curve behavior and prove regularity for generic S1-invariant J.
• Employs writhe bounds derived from higher asymptotics of pseudoholomorphic curves (Cristofaro-Gardiner-Hutchings-Zhang) to classify J-holomorphic buildings.
• Uses intersection theory arguments from Hutchings-Nelson (2016) to exclude higher-genus and multiple-cover contributions to the ECH differential.
• Computes the ECH index for generators using relative first Chern class, relative intersection pairing, and Conley-Zehnder index terms.
• Demonstrates that the ECH index defines a bijection to 2Z≥0 by analyzing lattice points in sublattices indexed by homology class Γ and line slopes of slope 2/(-e).

Experimental results

Research questions

• RQ1 How does the Z-grading on the ECH chain complex of a prequantization bundle over a Riemann surface refine the previously known Z2-grading isomorphism?
• RQ2 What is the precise structure of the ECH differential in this setting, and which J-holomorphic curves contribute to it?
• RQ3 Can the Morse-Bott direct limit argument be fully justified in this context, and what role does the isomorphism with Seiberg-Witten Floer cohomology play?
• RQ4 How can the classification of J-holomorphic buildings be achieved using writhe bounds and intersection theory?
• RQ5 Is the ECH index a bijection to 2Z≥0, and how does it match the exterior algebra grading on the base’s homology?

Key findings

• The ECH of a prequantization bundle over a closed Riemann surface is isomorphic as a Z-graded group to the exterior algebra of the homology of the base, with the grading matching the degree in the exterior algebra.
• The ECH index is shown to be a bijection from ECH generators to the non-negative even integers 2Z≥0, with the index increasing by exactly 2 when moving from one line of slope 2/(-e) to the next.
• The smallest ECH index on the line corresponding to M + (-e) is exactly 2 greater than the largest index on the line for M, confirming the correct grading filtration.
• The classification of J-holomorphic buildings excludes contributions from higher-genus curves and multiple covers of trivial cylinders, ensuring the differential only counts cylinders over Morse flow lines.
• The direct limit construction for filtered ECH is rigorously justified using the Hutchings-Taubes isomorphism, enabling the computation of the full ECH as a direct limit.
• The Conley-Zehnder index term ensures that generators on the same line of slope 2/(-e) have distinct indices increasing by 2 with each step in m+, preventing index collisions.