Lecture Notes on Embedded Contact Homology

Michael Hutchings

doi:10.1007/978-3-319-02036-5_9

Contact and Symplectic Topology pp 389-484 | Cite as

Lecture Notes on Embedded Contact Homology

Authors
Authors and affiliations

Michael Hutchings

Chapter

Part of the Bolyai Society Mathematical Studies book series (BSMS, volume 26)

Abstract

These notes give an introduction to embedded contact homology (ECH) of contact three-manifolds, gathering together many basic notions which are scattered across a number of papers. We also discuss the origins of ECH, including various remarks and examples which have not been previously published. Finally, we review the recent application to four-dimensional symplectic embedding problems. This article is based on lectures given in Budapest and Munich in the summer of 2012, a series of accompanying blog postings at floerhomology.wordpress.com , and related lectures at UC Berkeley in Fall 2012. There is already a brief introduction to ECH in the article of M. Hutchings (in Proceedings of the 2010 ICM, vol. II, pp. 1022–1041, 2010), but the present notes give much more background and detail.

Keywords

Contact Form Holomorphic Curve Floer Homology Holomorphic Curf Fredholm Index

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, to check access.

Appendix: Answers and Hints to Selected Exercises

1.4

We need to show that

$$ \lim_{k\to\infty}\frac{N(a,b)_k^2}{k} = 2ab. $$

(A.1)

Given nonnegative integers m and n, let T(m,n) denote the triangle in the plane bounded by the x and y axes and the line L through (m,n) with slope −b/a. Then N(a,b)_k=am+bn where T(m,n) encloses k+1 lattice points (including the edges). When k is large, the number of lattice points enclosed by T(m,n) is the area of the triangle, plus an O(k ^1/2) error. The line L intersects the axes at the points (a ⁻¹ N(a,b)_k,0) and (0,b ⁻¹ N(a,b)_k), so its area is

$$\operatorname {area}\bigl(T(m,n)\bigr)=\frac{N(a,b)_k^2}{2ab}. $$

Thus

$$k = \frac{N(a,b)_k^2}{2ab} + O\bigl(k^{1/2}\bigr). $$

This implies (A.1).

1.15

It is enough to show that

$$ 2\operatorname {vol}(X,\omega) = \operatorname {vol}(Y_+,\lambda_+) - \operatorname {vol}(Y_-, \lambda_-), $$

(A.2)

where $\operatorname {vol}(Y,\lambda) = \int_{Y}\lambda\wedge d \lambda$. To prove (A.2), let λ be a primitive of ω on X. Then by Stokes’s theorem,

$$2\operatorname {vol}(X,\omega) = \int_{Y_+}\lambda\wedge d\lambda_+ - \int_{Y_-}\lambda\wedge d\lambda_-. $$

Since dλ=dλ _± on Y _±, by Stokes’s theorem again we have

$$\int_{Y_\pm}\lambda\wedge d\lambda_\pm= \operatorname {vol}(Y_\pm,\lambda _\pm). $$

2.9

We use an infinitesimal analogue of the proof of Lemma 2.6. Let $\psi\in \operatorname {Ker}(D_{C})$. Let ε>0 be small and let C′ be the image of the map C→S ¹×Y _ϕ sending z↦exp_z(εψ(z)). Then

$$\int_{C'}\omega= \varepsilon ^2\int _C\omega(\partial_s\psi,\nabla _t \psi)ds\,dt + O\bigl(\varepsilon ^3\bigr). $$

Since C′ is homologous to C, we have ∫_C′ ω=0, so

$$ \int_C\omega(\partial_s\psi, \nabla_t\psi)ds\,dt = 0. $$

(A.3)

On the other hand, since $\psi\in \operatorname {Ker}(D_{C})$, we have ∇_t ψ=J∂ _s ψ, so the integrand above is

$$ \omega(\partial_s\psi,\nabla_t\psi) = \|\partial_s\psi\|^2, $$

(A.4)

where ∥⋅∥ denotes the metric on $T^{\operatorname {vert}}Y|_{\gamma}$ determined by ω and J. It follows from (A.3) and (A.4) that ∂ _s ψ≡0.

3.3

Given a Reeb orbit γ, the set of homotopy classes of trivializations of ξ|_τ is an affine space over ${\mathbb{Z}}$. For an appropriate sign convention, shifting the trivialization over $\gamma_{i}^{\pm}$ by 1 shifts c ₁ by ∓1 and shifts $CZ_{\tau}(\gamma_{i}^{\pm})$ by 2.

3.3

For an appropriate sign convention, shifting the trivialization τ over α _i by 1 shifts c ₁ by −m _i, shifts Q _τ by $m_{i}^{2}$, and shifts w _τ by −m _i(m _i−1).

3.11

Let T be the triangle in the plane which is bounded by the coordinate axes together with the line through (m ₁,m ₂) with slope −a/b, cf. the answer to Exercise 1.4. Then $\frac{1}{2}I(\alpha)$ can be interpreted as the number of lattice points in the triangle T (including the boundary) minus 1.

3.13

(a) Since the path $\varLambda_{\theta}^{+}(m)$ starts at the origin and stays below the line y=θx, the initial edge has slope less than θ. Since the path is the graph of a concave function, every subsequent edge also has slope less than θ. Thus b≤⌊aθ⌋. If b<⌊aθ⌋ then there is a lattice point which is above the path $\varLambda_{\theta}^{+}(m)$ but below the line y=θx, contradicting the definition of $\varLambda_{\theta}^{+}(m)$.

(b) Since the total vertical displacement of the path $\varLambda_{\theta}^{+}(m)$ is ⌊mθ⌋, it follows from part (a) that

$$\sum_{i=1}^k \lfloor q_i \theta \rfloor= \Biggl\lfloor \sum_{i=1}^kq_i \theta \Biggr\rfloor . $$

Since ⌊x⌋+⌊y⌋≤⌊x+y⌋ for any real numbers x,y, we have

$$\begin{aligned} \sum_{i\in I} \lfloor q_i\theta \rfloor &\le \biggl\lfloor \sum _{i\in I}q_i\theta \biggr\rfloor , \\ \sum_{i\in\{1,\ldots,k\}\setminus I} \lfloor q_i\theta \rfloor & \le \biggl\lfloor \sum_{i\in\{1,\ldots,k\}\setminus I}q_i\theta \biggr\rfloor . \end{aligned} $$

Adding the above two inequalities and comparing with the previous equation, we see that both inequalities must be equalities.

(c) Suppose that such proper subsets I,J exist. Let m ₁=∑_i∈I q _i=∑_j∈J r _j and let m ₂=m−m ₁. By part (b) applied to the subsets I, {1,…,k}∖I, and {1,…,k}, we have

$$\lfloor m_1\theta \rfloor + \lfloor m_2\theta \rfloor = \lfloor m\theta \rfloor. $$

By the analogue of part (b) for $p_{\theta}^{-}(m)$, we have

$$\lceil m_1\theta \rceil + \lceil m_2\theta \rceil = \lceil m\theta \rceil. $$

Subtracting the above two equations gives 2=1.

3.14

(a) Without loss of generality C is connected. Let a ₁,…,a _k denote the covering multiplicities of the positive ends of u, and let b ₁,…,b _l denote the covering multiplicities of the negative ends of u. Let g denote the genus of C. By the Fredholm index formula (3.1) and the Conley-Zehnder index formula (3.2), we have

$$\begin{aligned} \operatorname {ind}(u) &= 2g-2+k+l+ \sum _{i=1}^k \bigl(2 \lfloor a_i\theta \rfloor+1 \bigr) -\sum_{j=1}^l \bigl(2 \lfloor b_j\theta \rfloor +1 \bigr) \\ &= 2 \Biggl(g-1+\sum_{i=1}^k \lceil a_i\theta \rceil - \sum_{j=1}^l \lfloor b_j\theta \rfloor \Biggr) \\ &\ge2 \bigl(g-1+ \lceil m\theta \rceil- \lfloor m\theta \rfloor \bigr). \end{aligned} $$

Since ⌈mθ⌉−⌊mθ⌋=1, it follows that $\operatorname {ind}(u)\ge0$.

(b) We need to check: (i) if p≥q and q≥r then p≥r, and (ii) if p≥q and q≥p then p=q.

Suppose u ₁ is a branched cover with positive ends corresponding to p and negative ends corresponding to q, and u ₂ is a branched cover with positive ends corresponding to q and negative ends corresponding to r. Gluing these together gives a branched cover u ₁#u ₂ (defined up to sliding the branched points around) with positive ends corresponding to p and negative ends corresponding to r. It follows immediately from the index formula (3.1) that $\operatorname {ind}(u_{1}\#u_{2})= \operatorname {ind}(u_{1})+ \operatorname {ind}(u_{2})$. So if $\operatorname {ind}(u_{1})=\operatorname {ind}(u_{2})=0$, then $\operatorname {ind}(u_{1}\#u_{2})=0$ also, and this proves (i). Now suppose further that p=r. Then q=r, because otherwise u ₁#u ₂ has at least two branch points, so its domain has χ≤−2, so $\operatorname {ind}(u_{1}\#u_{2})\ge2$, a contradiction. This proves (ii).

(c) Let u be a connected genus 0 branched cover with positive ends corresponding to $p_{\theta}^{-}(m)$ and negative ends corresponding to $p_{\theta}^{+}(m)$. Write $p_{\theta}^{-}(m)=(a_{1},\ldots,a_{k})$ and $p_{\theta}^{+}(m)=(b_{1},\ldots,b_{l})$. By the calculation in part (a), we have

$$\operatorname {ind}(u) = 2 \Biggl(\sum_{i=1}^k \lceil a_i\theta \rceil - \sum_{j=1}^l \lfloor b_j\theta \rfloor - 1 \Biggr). $$

By Exercise 3.13(b) we have $\sum_{i=1}^{k} \lceil a_{i}\theta \rceil= \lceil m\theta \rceil$, and by symmetry $\sum_{j=1}^{l} \lfloor b_{j}\theta \rfloor = \lfloor m\theta \rfloor$. Hence $\operatorname {ind}(u)=0$.

(d) Suppose there exists a partition q with $p_{\theta}^{+}(m)>q$. Write $p_{\theta}^{+}(m)=(a_{1},\ldots,a_{k})$ and q=(b ₁,…,b _l). By Exercise 3.13(b) we have $\sum_{i=1}^{k} \lfloor a_{i}\theta \rfloor= \lfloor m\theta \rfloor$. By the calculation in part (a) above we have $\sum_{i=1}^{k} \lceil a_{i}\theta \rceil= \lceil m\theta \rceil$. These two equations imply that k=1. Thus the path $\varLambda_{\theta}^{+}(m)$ is just the line segment from (0,0) to (m,⌊mθ⌋).

Now the calculation in part (a) above also implies that $\sum_{j=1}^{l} \lfloor b_{j}\theta \rfloor= \lfloor m\theta \rfloor$. But this is impossible. To see why, order the numbers b _j so that ⌊b _j θ⌋/b _j≥⌊b _j+1 θ⌋/b _j+1. Let Λ′ be the path in the plane that starts at (0,0) and whose edge vectors are the segments (b _j,⌊b _j θ⌋) in order of increasing j. Since (b ₁,…,b _l)≠(m) and since there are no lattice points above the path $\varLambda_{\theta}^{+}(m)$ and below the line y=θx, it follows that the path Λ′ is below the path $\varLambda_{\theta}^{+}(m)$, with the two paths intersecting only at (0,0). Hence the right endpoint of Λ′ is below the right endpoint of $\varLambda_{\theta}^{+}(m)$, which means that ∑_j⌊b _j θ⌋<⌊mθ⌋.

By symmetry, there also does not exist a partition q with $q > p_{\theta}^{-}(m)$.

4.3

By Exercise 3.10 we have c _τ(C ₂)=1. Since $\operatorname {ind}(C_{2})=2$, it follows from (3.1) that

$$\chi(C_2) = CZ_\tau^{\operatorname {ind}}(C_2). $$

If ε is sufficiently small with respect to i, then $CZ_{\tau}(\gamma_{1}^{i})=2i-1$ when i>0, and $CZ_{\tau}(\gamma_{2}^{i-1})=2i-1$ when i>1. It follows that $CZ_{\tau}^{\operatorname {ind}}(C_{2})=0$ when i>1, and $CZ_{\tau}^{\operatorname {ind}}(C_{2})=1$ when i=1.

4.4

Without loss of generality, ${\mathcal{C}}_{0}=\emptyset$. We then compute that

$$CZ_\tau^{\operatorname {ind}}(C_2) = \left \{ \begin{array}{l@{\quad}l} i+j-1, & i>0,j>1,\\ i+1, & i>0, j=1,\\ j, & i=0, j>1,\\ 2, & i=0, j=1. \end{array} \right . $$

On the other hand, letting g denote the genus of C ₂, we have

$$\chi(C_2) = \left \{ \begin{array}{l@{\quad}l} -2g-i-j-1, & i>0, j>1,\\ -2g-i-1, & i>0, j=1,\\ -2g-j, & i=0, j>1,\\ -2g, & i=0, j=1. \end{array} \right . $$

Since c _τ(C ₂)=0 (by Exercise 3.10) and $\operatorname {ind}(C_{2})=2$, it follows from (3.1) that

$$\chi(C_2) = CZ_\tau^{\operatorname {ind}}(C_2)-2. $$

Combining the above three equations, we find that if i>0 or j>1 then g<0, which is a contradiction. Thus i=0 and j=1, and combining the above three equations again we find that g=0.

4.12

Otherwise g=1. Then Equation (4.9) (together with the fact that C ₁ has at least one positive end) implies that C ₁ has exactly one positive end at some hyperbolic orbit h _a,b, and all negative ends of C ₁ are elliptic. Let (a ₁,b ₁),…,(a _k,b _k) denote the vectors corresponding to the negative ends. The action of h _a,b is slightly less than $\sqrt{a^{2}+b^{2}}$, and the sum of the symplectic actions of the negative ends is slightly greater $\sum_{i=1}^{k}\sqrt{a_{i}^{2}+b_{i}^{2}}$. Since the differential decreases symplectic action,

$$\sum_{i=1}^k\sqrt{a_i^2+b_i^2} < \sqrt{a^2+b^2}. $$

But this contradicts the triangle inequality, since $\sum_{i=1}^{k}(a_{i},b_{i})=(a,b)$, since h _a,b is homologous in T ³ to $\sum_{i=1}^{k} e_{a_{i},b_{i}}$.

4.13

Let Λ be any polygonal path with edge vectors v ₁,…,v _k. Then

$$\ell_\varOmega(\varLambda) = \sum_{i=1}^k \langle v_i,w_i\rangle $$

where w _i∈∂Ω′ is a point at which an outward normal vector to Ω′ is a positive multiple of v _i. (When w _i is a corner of ∂Ω′, “an outward normal vector” means a vector whose direction is between the directions of the limits of the normal vectors on either side of w _i.) If we replace Ω′ by its translate by some vector η, then the above formula is replaced by

$$\ell_\varOmega(\varLambda) = \sum_{i=1}^k \langle v_i,w_i+\eta\rangle. $$

If Λ is a loop, then the two formulas for ℓ _Ω(Λ) agree since ∑_i v _i=0.

5.9

By the relative adjunction formula (3.3), and since equality holds in the writhe bound (3.9), we have

$$-\chi(C) = -c_\tau(C) + Q_\tau(C) + CZ_\tau^I(C) - CZ_\tau^{\operatorname {ind}}(C). $$

So by the definition of J ₀ in (5.13) and (5.14), we need to show that

$$\sum_i\bigl(n_i^+-1\bigr) + \sum _j\bigl(n_j^--1\bigr) = CZ_\tau^{J_0}(C) - CZ_\tau^I(C) + CZ_\tau^{\operatorname {ind}}(C). $$

This equation can be proved one Reeb orbit at a time. Namely, it is enough to show that for each i, if C has positive ends at covers of α _i with multiplicities $q_{1},\ldots,q_{n_{i}^{+}}$ where $\sum_{k=1}^{n_{i}^{+}}q_{k}=m_{i}$, then

$$ n_i^+-1 = -CZ_\tau\bigl( \alpha_i^{m_i}\bigr) + \sum_{k=1}^{n_i^+}CZ_\tau \bigl(\alpha^{q_k}\bigr), $$

(A.5)

and an analogous equation for each Reeb orbit β _j.

To prove (A.5), first note that if α _i is hyperbolic, then $n_{i}^{+}=m_{i}=1$ and the equation is trivial. Suppose now that α _i is elliptic with rotation angle θ with respect to τ. Then (A.5) becomes

$$0 = -2 \lfloor m_i\theta \rfloor + \sum _{k=1}^{n_i^+}2 \lfloor q_k\theta \rfloor. $$

This last equation holds by the partition conditions and Exercise 3.13(b).

References

[1]
P. Biran, From symplectic packing to algebraic geometry and back, in European Congress of Mathematicians, vol. II. Progress in Math, vol. 202 (Birkhäuser, Basel, 2000), pp. 507–524 Google Scholar
[2]
F. Bourgeois, A Morse-Bott approach to contact homology, in Symplectic and Contact Topology: Interactions and Perspectives, Toronto, 2001. Fields Inst. Commun., vol. 35 (AMS, Providence, 2003), pp. 55–77 Google Scholar
[3]
F. Bourgeois, K. Mohnke, Coherent orientations in symplectic field theory. Math. Z. 248, 123–146 (2004) CrossRefMATHMathSciNetGoogle Scholar
[4]
F. Bourgeois, Y. Eliashberg, H. Hofer, K. Wysocki, E. Zehnder, Compactness results in symplectic field theory. Geom. Topol. 7, 799–888 (2003) CrossRefMATHMathSciNetGoogle Scholar
[5]
K. Cieliebak, H. Hofer, J. Latschev, F. Schlenk, Quantitative symplectic geometry, in Dynamics, Ergodic Theory, and Geometry. MSRI Publ, vol. 54 (Cambridge University Press, Cambridge, 2007), pp. 1–44 CrossRefGoogle Scholar
[6]
V. Colin, K. Honda, Reeb vector fields and open book decompositions. J. Eur. Math. Soc. 15, 443–507 (2013) CrossRefMATHMathSciNetGoogle Scholar
[7]
V. Colin, P. Ghiggini, K. Honda, M. Hutchings, Sutures and contact homology I. Geom. Topol. 15, 1749–1842 (2011) CrossRefMATHMathSciNetGoogle Scholar
[8]
V. Colin, P. Ghiggini, K. Honda, HF=ECH via open book decompositions: a summary. arXiv:1103.1290
[9]
D. Cristofaro-Gardiner, The absolute gradings in embedded contact homology and Seiberg-Witten Floer cohomology. Algebr. Geom. Topol. 13, 2239–2260 (2013) CrossRefMATHMathSciNetGoogle Scholar
[10]
D. Cristofaro-Gardiner, M. Hutchings, From one Reeb orbit to two. arXiv:1202.4839
[11]
D. Cristofaro-Gardiner, V. Gripp, M. Hutchings, The asymptotics of ECH capacities. arXiv:1210.2167
[12]
D. Dragnev, Fredholm theory and transversality for noncompact pseudoholomorphic maps in symplectizations. Commun. Pure Appl. Math. 57, 726–763 (2004) CrossRefMATHMathSciNetGoogle Scholar
[13]
I. Ekeland, H. Hofer, Symplectic topology and Hamiltonian dynamics. II. Math. Z. 203, 553–567 (1990) CrossRefMATHMathSciNetGoogle Scholar
[14]
Y. Eliashberg, A. Givental, H. Hofer, Introduction to symplectic field theory. Geom. Funct. Anal. (2000), Special Volume, Part II, 560–673 Google Scholar
[15]
O. Fabert, Contact homology of Hamiltonian mapping tori. Comment. Math. Helv. 85, 203–241 (2010) CrossRefMATHMathSciNetGoogle Scholar
[16]
D. Farris, The embedded contact homology of nontrivial circle bundles over Riemann surfaces. UC Berkeley PhD thesis, 2011 Google Scholar
[17]
D. Frenkel, D. Müller, Symplectic embeddings of 4-dimensional ellipsoids into cubes. arXiv:1210.2266
[18]
H. Hofer, K. Wysocki, E. Zehnder, Properties of pseudo-holomorphic curves in symplectisations. II. Embedding controls and algebraic invariants. Geom. Funct. Anal. 5, 270–328 (1995) CrossRefMATHMathSciNetGoogle Scholar
[19]
H. Hofer, K. Wysocki, E. Zehnder, Properties of pseudoholomorphic curves in symplectisations. I. Asymptotics. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 13, 337–379 (1996) MATHMathSciNetGoogle Scholar
[20]
H. Hofer, K. Wysocki, E. Zehnder, A general Fredholm theory. I. A splicing-based differential geometry. J. Eur. Math. Soc. 9, 841–876 (2007) CrossRefMATHMathSciNetGoogle Scholar
[21]
M. Hutchings, An index inequality for embedded pseudoholomorphic curves in symplectizations. J. Eur. Math. Soc. 4, 313–361 (2002) CrossRefMATHMathSciNetGoogle Scholar
[22]
M. Hutchings, The embedded contact homology index revisited, in New Perspectives and Challenges in Symplectic Field Theory. CRM Proc. Lecture Notes, vol. 49 (Amer. Math. Soc., Providence, 2009), pp. 263–297 Google Scholar
[23]
M. Hutchings, Embedded contact homology and its applications, in Proceedings of the 2010 ICM, vol. II (2010), pp. 1022–1041 Google Scholar
[24]
M. Hutchings, Quantitative embedded contact homology. J. Differ. Geom. 88, 231–266 (2011) MATHMathSciNetGoogle Scholar
[25]
M. Hutchings, Recent progress on symplectic embedding problems in four dimensions. Proc. Natl. Acad. Sci. USA 108, 8093–8099 (2011) CrossRefMATHMathSciNetGoogle Scholar
[26]
M. Hutchings, Embedded contact homology as a (symplectic) field theory, in preparation Google Scholar
[27]
M. Hutchings, M. Sullivan, The periodic Floer homology of a Dehn twist. Algebr. Geom. Topol. 5, 301–354 (2005) CrossRefMATHMathSciNetGoogle Scholar
[28]
M. Hutchings, M. Sullivan, Rounding corners of polygons and the embedded contact homology of T ³. Geom. Topol. 10, 169–266 (2006) CrossRefMATHMathSciNetGoogle Scholar
[29]
M. Hutchings, C.H. Taubes, Gluing pseudoholomorphic curves along branched covered cylinders I. J. Symplectic Geom. 5, 43–137 (2007) CrossRefMATHMathSciNetGoogle Scholar
[30]
M. Hutchings, C.H. Taubes, Gluing pseudoholomorphic curves along branched covered cylinders II. J. Symplectic Geom. 7, 29–133 (2009) CrossRefMATHMathSciNetGoogle Scholar
[31]
M. Hutchings, C.H. Taubes, The Weinstein conjecture for stable Hamiltonian structures. Geom. Topol. 13, 901–941 (2009) CrossRefMATHMathSciNetGoogle Scholar
[32]
M. Hutchings, C.H. Taubes, Proof of the Arnold chord conjecture in three dimensions II. Geom. Topol. 17, 2601–2688 (2013) CrossRefMATHGoogle Scholar
[33]
E. Ionel, T. Parker, The Gromov invariants of Ruan-Tian and Taubes. Math. Res. Lett. 4, 521–532 (1997) CrossRefMATHMathSciNetGoogle Scholar
[34]
P.B. Kronheimer, T.S. Mrowka, Monopoles and Three-Manifolds (Cambridge University Press, Cambridge, 2008) Google Scholar
[35]
C. Kutluhan, Y.-J. Lee, C. Taubes, HF=HM I: Heegaard Floer homology and Seiberg-Witten Floer homology. arXiv:1007.1979
[36]
C. Kutluhan, Y.-H. Lee, C. Taubes, HF=HM II: Reeb orbits and holomorphic curves for the ech/Heegaard-Floer correspondence. arXiv:1008.1595
[37]
J. Latschev, C. Wendl, Algebraic torsion in contact manifolds. Geom. Funct. Anal. 21, 1144–1195 (2011), corrected version of appendix at arXiv:1009.3262 CrossRefMATHMathSciNetGoogle Scholar
[38]
Y.-J. Lee, C. Taubes, Periodic Floer homology and Seiberg-Witten Floer cohomology. J. Symplectic Geom. 10, 81–164 (2012) CrossRefMATHMathSciNetGoogle Scholar
[39]
T.-J. Li, A.-K. Liu, The equivalence between SW and Gr in the case where b ⁺=1. Int. Math. Res. Not. 7, 335–345 (1999) CrossRefGoogle Scholar
[40]
D. McDuff, From symplectic deformation to isotopy, in Topics in Symplectic 4-Manifolds, Irvine, CA, 1996. First Int. Press Lect. Ser. I (International Press, Cambridge, 1996), pp. 85–99 Google Scholar
[41]
D. McDuff, The Hofer conjecture on embedding symplectic ellipsoids. J. Differ. Geom. 88, 519–532 (2011) MATHMathSciNetGoogle Scholar
[42]
D. McDuff, D. Salamon, J-Holomorphic Curves and Symplectic Topology. AMS Colloquium Publications, vol. 52 Google Scholar
[43]
D. McDuff, F. Schlenk, The embedding capacity of 4-dimensional symplectic ellipsoids. Ann. Math. 175, 1191–1282 (2012) CrossRefMATHMathSciNetGoogle Scholar
[44]
G. Meng, C.H. Taubes, $\underline{\mathit{SW}}=\mbox{Milnor torsion}$. Math. Res. Lett. 3, 661–674 (1996) CrossRefMATHMathSciNetGoogle Scholar
[45]
J. Morgan, The Seiberg-Witten Equations and Applications to the Topology of Smooth Four-Manifolds. Mathematical Notes, vol. 44 (Princeton Univ. Press, Princeton, 1996) MATHGoogle Scholar
[46]
P. Ozsváth, Z. Szabó, Holomorphic disks and topological invariants for closed three-manifolds. Ann. Math. 159, 1027–1158 (2004) CrossRefMATHGoogle Scholar
[47]
B. Parker, Holomorphic curves in Lagrangian torus fibrations. Stanford University PhD thesis, 2005 Google Scholar
[48]
D. Salamon, Lectures on Floer homology, in Symplectic Geometry and Topology, Park City, UT, 1997. IAS/Park City Math. Ser., vol. 7 (AMS, Providence, 1999) Google Scholar
[49]
D. Salamon, Seiberg-Witten invariants of mapping tori, symplectic fixed points, and Lefschetz numbers. Turk. J. Math. 23, 117–143 (1999). 6th Gökova Geometry-Topology Conference MATHMathSciNetGoogle Scholar
[50]
M. Schwarz, Cohomology operations from S ¹-cobordisms in Floer homology. PhD thesis, ETH Zürich, 1995 Google Scholar
[51]
R. Siefring, Relative asymptotic behavior of pseudoholomorphic half-cylinders. Commun. Pure Appl. Math. 61, 1631–1684 (2008) CrossRefMATHMathSciNetGoogle Scholar
[52]
R. Siefring, Intersection theory of punctured pseudoholomorphic curves. Geom. Topol. 15, 2351–2457 (2011) CrossRefMATHMathSciNetGoogle Scholar
[53]
C.H. Taubes, Counting pseudo-holomorphic submanifolds in dimension four. J. Differ. Geom. 44, 818–893 (1996) MATHMathSciNetGoogle Scholar
[54]
C.H. Taubes, The structure of pseudoholomorphic subvarieties for a degenerate almost complex structure and symplectic form on S ¹×B ³. Geom. Topol. 2, 221–332 (1998) CrossRefMATHMathSciNetGoogle Scholar
[55]
C.H. Taubes, Seiberg-Witten and Gromov Invariants for Symplectic 4-Manifolds. First International Press Lecture Series, vol. 2 (International Press, Cambridge, 2000) MATHGoogle Scholar
[56]
C.H. Taubes, Pseudolomorphic punctured spheres in ${\mathbb{R}}\times(S^{1}\times S^{2})$: properties and existence. Geom. Topol. 10, 785–928 (2006) CrossRefMATHMathSciNetGoogle Scholar
[57]
C.H. Taubes, The Seiberg-Witten equations and the Weinstein conjecture. Geom. Topol. 11, 2117–2202 (2007) CrossRefMATHMathSciNetGoogle Scholar
[58]
C.H. Taubes, Embedded contact homology and Seiberg-Witten Floer homology I. Geom. Topol. 14, 2497–2581 (2010) CrossRefMATHMathSciNetGoogle Scholar
[59]
C.H. Taubes, Embedded contact homology and Seiberg-Witten Floer homology V. Geom. Topol. 14, 2961–3000 (2010) CrossRefMATHMathSciNetGoogle Scholar
[60]
V. Turaev, A combinatorial formulation for the Seiberg-Witten invariants of 3-manifolds. Math. Res. Lett. 5, 583–598 (1998) CrossRefMATHMathSciNetGoogle Scholar
[61]
C. Wendl, Compactness for embedded pseudoholomorphic curves in 3-manifolds. J. Eur. Math. Soc. 12, 313–342 (2010) CrossRefMATHMathSciNetGoogle Scholar
[62]
C. Wendl, Automatic transversality and orbifolds of punctured holomorphic curves in dimension four. Comment. Math. Helv. 85, 347–407 (2010) CrossRefMATHMathSciNetGoogle Scholar
[63]
M.-L. Yau, Vanishing of the contact homology of overtwisted contact 3-manifolds. Bull. Inst. Math. Acad. Sin. 1, 211–229 (2006), with an appendix by Y. Eliashberg ADSMATHMathSciNetGoogle Scholar

Copyright information

Authors and Affiliations

Michael Hutchings
- 1

1.University of CaliforniaBerkeleyUSA

Lecture Notes on Embedded Contact Homology

Abstract

Keywords

References

Copyright information

Authors and Affiliations

Personalised recommendations

Cite chapter