## Involve: A Journal of Mathematics

• Involve
• Volume 10, Number 3 (2017), 417-442.

### Reeb dynamics of the link of the $A_n$ singularity

#### Abstract

The link of the $An$ singularity, $LAn ⊂ ℂ3$ admits a natural contact structure $ξ0$ coming from the set of complex tangencies. The canonical contact form $α0$ associated to $ξ0$ is degenerate and thus has no isolated Reeb orbits. We show that there is a nondegenerate contact form for a contact structure equivalent to $ξ0$ that has two isolated simple periodic Reeb orbits. We compute the Conley–Zehnder index of these simple orbits and their iterates. From these calculations we compute the positive $S1$-equivariant symplectic homology groups for $(LAn,ξ0)$. In addition, we prove that $(LAn,ξ0)$ is contactomorphic to the lens space $L(n + 1,n)$, equipped with its canonical contact structure $ξstd$.

#### Article information

Source
Involve, Volume 10, Number 3 (2017), 417-442.

Dates
Revised: 7 June 2016
Accepted: 9 June 2016
First available in Project Euclid: 12 December 2017

https://projecteuclid.org/euclid.involve/1513087847

Digital Object Identifier
doi:10.2140/involve.2017.10.417

Mathematical Reviews number (MathSciNet)
MR3583874

Zentralblatt MATH identifier
1379.37032

#### Citation

Abbrescia, Leonardo; Huq-Kuruvilla, Irit; Nelson, Jo; Sultani, Nawaz. Reeb dynamics of the link of the $A_n$ singularity. Involve 10 (2017), no. 3, 417--442. doi:10.2140/involve.2017.10.417. https://projecteuclid.org/euclid.involve/1513087847

#### References

• F. Bourgeois and A. Oancea, “$S^1$-equivariant symplectic homology and linearized contact homology”, preprint, 2012. To appear in Int. Math. Res. Not.
• F. Bourgeois and A. Oancea, “Erratum to:\! An exact sequence for contact- and symplectic homology”, Invent. Math. 200:3 (2015), 1065–1076.
• E. Brieskorn, “Beispiele zur Differentialtopologie von Singularitäten”, Invent. Math. 2 (1966), 1–14.
• H. Geiges, An introduction to contact topology, Cambridge Studies in Advanced Mathematics 109, Cambridge University Press, 2008.
• J. Gutt, “Generalized Conley–Zehnder index”, Ann. Fac. Sci. Toulouse Math. $(6)$ 23:4 (2014), 907–932.
• J. Gutt, “The positive equivariant symplectic homology as an invariant for some contact manifolds”, preprint, 2015.
• M. Hutchings and J. Nelson, “Cylindrical contact homology for dynamically convex contact forms in three dimensions”, preprint, 2014. To appear in J. Symplectic Geom.
• M. Hutchings and J. Nelson, “Invariance and an integral lift of cylindrical contact homology for dynamically convex contact forms”, in preparation.
• Y. Ito and M. Reid, “The McKay correspondence for finite subgroups of ${\rm SL}(3,\mathbb{C})$”, pp. 221–240 in Higher-dimensional complex varieties (Trento, 1994), edited by M. Andreatta and T. Peternell, de Gruyter, Berlin, 1996.
• A. Keating, “Homological mirror symmetry for hypersurface cusp singularities”, preprint, 2015.
• M. Kwon and O. van Koert, “Brieskorn manifolds in contact topology”, Bull. Lond. Math. Soc. 48:2 (2016), 173–241.
• D. McDuff and D. Salamon, Introduction to symplectic topology, 2nd ed., Oxford University Press, 1998.
• M. McLean, “Reeb orbits and the minimal discrepancy of an isolated singularity”, Invent. Math. 204:2 (2016), 505–594.
• M. McLean and A. Ritter, “The cohomological McKay correspondence and symplectic homology”, in preparation.
• J. Milnor, Singular points of complex hypersurfaces, Annals of Mathematics Studies 61, Princeton University Press, 1968.
• E. E. Moise, “Affine structures in $3$-manifolds, V: The triangulation theorem and Hauptvermutung”, Ann. Math. 56:2 (1952), 96–114.
• J. Nelson, “Automatic transversality in contact homology, I: Regularity”, Abh. Math. Semin. Univ. Hambg. 85:2 (2015), 125–179.
• J. Nelson, “Automatic transversality in contact homology, II: Invariance and computations”, in preparation.
• A. F. Ritter, “Deformations of symplectic cohomology and exact Lagrangians in ALE spaces”, Geom. Funct. Anal. 20:3 (2010), 779–816.
• J. Robbin and D. Salamon, “The Maslov index for paths”, Topology 32:4 (1993), 827–844.
• P. Seidel, “A biased view of symplectic cohomology”, pp. 211–253 in Current developments in mathematics, 2006, edited by B. Mazur et al., Int. Press, Somerville, MA, 2008.
• P. Seidel, Fukaya categories and Picard–Lefschetz theory, European Mathematical Society, Zürich, 2008.
• I. Ustilovsky, “Infinitely many contact structures on $S^{4m+1}$”, Internat. Math. Res. Notices 14 (1999), 781–791.