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On fillings of contact links of quotient singularities

Zhengyi Zhou

Abstract

We study several aspects of fillings for links of general quotient singularities using Floer theory, including co-fillings, Weinstein fillings, strong fillings, exact fillings and exact orbifold fillings, focusing on non-existence of exact fillings of contact links of isolated terminal quotient singularities. We provide an extensive list of isolated terminal quotient singularities whose contact links are not exactly fillable, including $\mathbb{C}^n/(\mathbb{Z}/2)$ for $n\ge 3$, which settles a conjecture of Eliashberg, quotient singularities from general cyclic group actions and finite subgroups of $SU(2)$, and all terminal quotient singularities in complex dimension $3$. We also obtain uniqueness of the orbifold diffeomorphism type of exact orbifold fillings of contact links of some isolated terminal quotient singularities.

On fillings of contact links of quotient singularities

Abstract

We study several aspects of fillings for links of general quotient singularities using Floer theory, including co-fillings, Weinstein fillings, strong fillings, exact fillings and exact orbifold fillings, focusing on non-existence of exact fillings of contact links of isolated terminal quotient singularities. We provide an extensive list of isolated terminal quotient singularities whose contact links are not exactly fillable, including for , which settles a conjecture of Eliashberg, quotient singularities from general cyclic group actions and finite subgroups of , and all terminal quotient singularities in complex dimension . We also obtain uniqueness of the orbifold diffeomorphism type of exact orbifold fillings of contact links of some isolated terminal quotient singularities.
Paper Structure (36 sections, 54 theorems, 106 equations, 2 figures)

This paper contains 36 sections, 54 theorems, 106 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Fillings of Contact links of isolated quotient singularities
  3. Weinstein fillings, strong fillings, and co-fillings.
  4. Non-existence of exact fillings
  5. Uniqueness of the diffeomorphism type of exact orbifold fillings
  6. Basics of the contact link $(S^{2n-1}/G,\xi_{\rm std})$
  7. Topology of $S^{2n-1}/G$
  8. Reeb dynamics on $S^{2n-1}/G$
  9. Chen-Ruan cohomology of ${\mathbb C}^n/G$
  10. Product on Chen-Ruan cohomology
  11. Coproduct on Chen-Ruan cohomology
  12. Minimal discrepancy number of ${\mathbb C}^n/G$
  13. Symplectic cohomology of fillings of $(S^{2n-1}/G,\xi_{\rm std})$
  14. Symplectic cohomology
  15. Symplectic cohomology of exact fillings of $(S^{2n-1}/G,\xi_{\rm std})$
  16. ...and 21 more sections

Key Result

Theorem 2

For $n\ge 3$, $(\mathbb{R}\mathbb{P}^{2n-1},\xi_{\rm std})$ is not exactly fillable.

Figures (2)

  • Figure 2: Secondary coproduct $\bm{\lambda}$, numbers are weights at the puncture
  • Figure 3: Pictorial proof of \ref{['thm:EGL']}

Theorems & Definitions (118)

  • Conjecture 1: question
  • Theorem 2
  • Definition 1.1
  • Remark 1.2
  • Theorem 3
  • Remark 1.3
  • Proposition 4
  • Conjecture 5
  • Theorem 6
  • Theorem 7
  • ...and 108 more