Erratum to: An exact sequence for contact and symplectic homology
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1 Erratum to: Invent. Math. 175, 611–680 (2009) DOI: 10.1007/s0022200801591
The goal of this note is to make two corrections to the original article. The first correction concerns the transversality assumptions for the definition of linearized contact homology. The second correction concerns the description of the connecting map \(D\) in terms of the contact complex. These corrections are described, respectively, in Sects. 1 and 2 below.
The equations in the original article are numbered by \(1,2,3,\ldots \). The equations in the current note are numbered by \(x.1, x.2, x.3,\dots \) where \(x\) is the number of the section.
2 Transversality assumptions
In this section we correct conditions \((A)\) and \((B_a)\) in Remark 9, p. 633 of the original article by formulating below the stronger conditions \((\widetilde{A})\) and \((\widetilde{B}_a)\) under which linearized contact homology is defined in a given free homotopy class \(a\). As stated in Remark 9, p. 633 of the original article we expect the results of the original article to hold in complete generality once a suitable perturbation scheme has been worked out (for example the polyfold theory of Hofer et al. [5]), since our arguments are essentially independent of the details of such a scheme.
For the reader’s convenience we recall the notation from the original article: \((M,\xi )\) is a contact manifold with symplectic filling \((W,\omega )\), such that \(\omega \) admits a primitive near the boundary which restricts to a positive contact form \(\lambda \) on the boundary; the symplectic completion of \((W,\omega )\) is denoted \((\widehat{W},\widehat{\omega })\); \(J\) is a compatible almost complex structure on \(\widehat{W}\) which is cylindrical on the symplectization part of \(\widehat{W}\); \(J_\infty \) is the corresponding compatible almost complex structure on the symplectization \({\mathbb {R}}\times M\); \(a\) denotes a free homotopy class of loops in \(W\), \(i:M\hookrightarrow W\) is the inclusion, and \(i^{1}(a)\) denotes the set of free homotopy classes of loops in \(M\) which are mapped to \(a\) via the inclusion; \({\mathcal {P}}^{i^{1}(a)}_\lambda \) is the set of periodic Reeb orbits of \(\lambda \) in \(i^{1}(a)\).
In Remark 9, p. 633 of the original article we formulated conditions \((A)\) and \((B_a)\) under which linearized contact homology \(HC_*^{i^{1}(a)}(\lambda , J)\) was supposed to be welldefined. It was brought to our attention by M. Abouzaid, J. Latschev and J. Nelson that condition \((B_a)\) is not strong enough as shown by the following example. Consider the ellipsoid \(E:=\{(z_1,z_2)\in {\mathbb {C}}^2\, : \, z_1^2/a_1^2+z_2^2/a_2^2=1\}\) with \(0<\sqrt{2}a_1<a_2\) and \(a_1,a_2\) rationally independent. The contact form is induced by \(\frac{1}{2}(\sum x_idy_i  y_idx_i)\) on \({\mathbb {R}}^4\equiv {\mathbb {C}}^2\), and the closed Reeb orbits are iterates \(\gamma ^i_k\), \(i=1,2\), \(k\ge 1\) of the simple orbits \(\gamma ^1:=E\cap \{z_2=0\}\), \(\gamma ^2:=E\cap \{z_1=0\}\). Since \(a_1,a_2\) are rationally independent these orbits are nondegenerate, and since \(\sqrt{2}a_1<a_2\) the Conley–Zehnder indices of \(\gamma ^1_1,\gamma ^1_2\) are \(\mu (\gamma ^1_1)=3\), \(\mu (\gamma ^1_2)=5\) and their grading is \(\gamma ^1_1=2\), \(\gamma ^1_2=4\). Let \(J\) be some generic almost complex structure as above. The moduli space \({\mathcal {M}}(\gamma ^1_1,\emptyset ;J)\) (cf. p. 630 of the original article) is nonempty and has dimension \(2\) (the asymptote \(\gamma ^1_1\) is simple and transversality can be achieved for generic \(J\)). On the other hand, the moduli space \({\mathcal {M}}(\gamma ^1_2,\gamma ^1_1,\gamma ^1_1;J_\infty )\) (cf. p. 629 of the original article) has virtual dimension \(0\) and is always nonempty, since it contains the double branched covers of the trivial cylinder over \(\gamma ^1_1\), which form a \(2\)dimensional family. Thus transversality can never be achieved for this moduli space. The point now is that pairs \([u,v]\in {\mathcal {M}}_c(\gamma ^1_2,\gamma ^1_1;J)\) with \(u\in {\mathcal {M}}(\gamma ^1_2,\gamma ^1_1,\gamma ^1_1;J_\infty )\) and \(v\in {\mathcal {M}}(\gamma ^1_1,\emptyset ;J)\) fall outside the scope of assumption \((B_a)\) in Remark 9, p. 633 of the original article since \(\dim \,{\mathcal {M}}(\gamma ^1_1,\emptyset ;J)>0\). On the other hand, nothing prevents a priori such a pair to appear on the boundary of \({\mathcal {M}}(\gamma ^1_2,\gamma ^1_1;J_\infty )\), which would destroy the relation \({\partial }^2=0\) for the differential in linearized contact homology.
Condition \((A)\) is not sufficient either as shown by the following idealized example: an index 2 cylinder in the symplectization could break into an index 1 pair of pants and an index 1 plane in the symplectization, which can be viewed in \(\widehat{W}\). In order to analyze the contribution of this configuration one needs to ensure regularity of index 1 planes in \(\widehat{W}\).
Correction for conditions \((A)\) and \((B_a)\) of Remark 9, p. 633 of the original article
 (\(\widetilde{A}\))

\(J\) is regular for holomorphic planes in \(\widehat{W}\) which belong to moduli spaces \({\mathcal {M}}(\gamma ',\emptyset ;J)\) of virtual dimension \(\le \!{1}\);
 (\(\widetilde{B}_a\))

\(J_\infty \) is regular for punctured holomorphic cylinders asymptotic at \(\pm \infty \) to closed Reeb orbits in \(i^{1}(a)\), belonging to moduli spaces of virtual dimension \(\le \!2\), and asymptotic at the punctures to elements \(\gamma '\in {\mathcal {P}}^{i^{1}(0)}(\lambda )\) such that there exists a \(J\)holomorphic building of type \(01k_+\), \(k_+\ge 0\) in the sense of [2, §8.1] with exactly one positive puncture and asymptote \(\gamma '\). (By definition, a building of type \(01k_+\) has 1 level in \(\widehat{W}\) and \(k_+\) levels in \({\mathbb {R}}\times M\).)
We wish to stress that assumptions \((\widetilde{A})\) and \((\widetilde{B}_a)\) depend on \(\lambda \) and \(J\). In no way do they suffice to prove invariance of \(HC_*^{i^{1}(a)}(\lambda ,J)\) with respect to deformations of \(\lambda \) or \(J\). As explicitly stated in Remark 7, p. 633 of the original article the invariance with respect to \(\lambda \) and \(J\) needs the polyfold formalism currently being developed by Hofer et al. [5]. Alternatively, invariance follows from the isomorphism between \(HC_*^{i^{1}(a)}(\lambda ,J)\) and positive \(S^1\)equivariant symplectic homology \(SH_*^{S^1,+}(W,\omega )\) with \({\mathbb {Q}}\)coefficients [4]. The proofs of the original article are written for a specific choice of \(\lambda \) and \(J\) that obey the assumptions above.
We give below the proof that \(HC_*^{i^{1}(a)}(\lambda ,J)\) is defined under assumptions \((\widetilde{A})\) and \((\widetilde{B}_a)\). Along the way, we need to correct the Eqs. (36) and (77) in the original article.
Correction for Eq. (36)
To prove the claim we again appeal to the SFT compactness theorem [2], by which the boundary of the \(1\)dimensional moduli space \({\mathcal {M}}^A_c({\overline{\gamma }}',{\underline{\gamma }}';J)/{\mathbb {R}}\) containing a \(1\)parameter family of punctured cylinders of index \(2\) in \({\mathbb {R}}\times M\) capped by rigid holomorphic planes in \(\widehat{W}\) consists of holomorphic buildings of type \(01k_+\) with \(k_+ \ge 2\). We prove that we have \(k_+=2\). Indeed, assumption \((\widetilde{B}_a)\) implies that the index of all levels in the symplectization is \(\ge 1\), and assumption \((\widetilde{A})\) implies that the index of all levels in \(\widehat{W}\) is \(\ge 0\). Since the total index is \(2\) we obtain that \(k_+=2\).
The first line of (1.2) describes the situation in which \({\overline{\gamma }}'\), \({\underline{\gamma }}'\) are asymptotes of punctured cylinders which lie on different levels in the symplectization. The second and third lines of (1.2) describe the situation in which \({\overline{\gamma }}'\), \({\underline{\gamma }}'\) are the two asymptotes at \(\pm \infty \) of a punctured cylinder of index \(1\) in the symplectization. This punctured cylinder is capped by rigid holomorphic planes in \(\widehat{W}\) at all punctures but one, where it is capped by an index \(1\) holomorphic building of type \(011\). The latter is described by the third line in (1.2). \(\square \)
Correction for Eq. (77)
Correction for the discussion of the examples in Remark 9, p. 634 of the original article
In Examples (i) and (ii) the stronger assumption \((\widetilde{B}_0)\) is violated, as we explain below. In Example (iii) we need to strengthen the condition \(\dim \, L\ge 4\) to \(\dim \, L\ge 5\), so that \((\widetilde{A})\) becomes vacuous; the rest of the discussion holds verbatim to show that \((\widetilde{B}_a)\) is verified.
Example (i). As in p. 633 of the original article one sees that condition \((\widetilde{A})\) is satisfied. However, assumption \((\widetilde{B}_0)\) is never satisfied. To see this, one can refer to the example of the ellipsoid \(E=\{(z_1,z_2)\in {\mathbb {C}}^2\, : \, z_1^2/a_1^2+z_2^2/a_2^2=1\}\) with \(a_1\), \(a_2\) rationally independent and \(\sqrt{2}a_1<a_2\) presented at the beginning of Sect. 1. In that example we exhibited an index \(0\) pair of pants in the symplectization, capped at one of the punctures by a plane of index \(2\). In higher dimensions the ellipsoid \(E=\{(z_1,\dots ,z_n)\in {\mathbb {C}}^n\, : \, \sum _{j=1}^n z_j^2/a_j^2=1\}\) with \(a_1,\dots ,a_n\) rationally independent and \(\sqrt{2}a_1<a_2<\dots <a_n\) features similar properties: one can exhibit a pair of pants of negative index \(2n+4\) capped at one of the punctures by a plane of positive index \(2n2\). Subcritical Stein manifolds are obtained by attaching subcritical Weinstein handles; these always contain such contact ellipsoids which violate condition \((\widetilde{B}_0)\).
The paragraph that follows is conjectural: MeiLin Yau claims in [10] that, for a special choice of almost complex structure, the cylindrical contact homology for boundaries of subcritical Stein manifolds with vanishing first Chern class can be expressed by an explicit formula in terms of the singular homology of the filling relative to its boundary. The proof in [10] has a gap. Specifically, there is a problem in the proof of [10, Lemma7.6] which makes use of a flawed covering trick, namely the contact complex for the new contact form as \(k\rightarrow \infty \) contains potentially many more generators than the ones that are considered in [10, §7], and these new generators are not accounted for in [10, §7]. However, we believe that the main formula in [10] is correct. Then Proposition 9 is to be read as expressing the fact that a long exact sequence as the one in Theorem 1 is compatible with this computation.
Example (ii). Given an upper bound \(\alpha \) on the action, we choose a Morse–Bott perturbation \((\widehat{\omega }_\epsilon ,J_\epsilon )\) with \(\epsilon >0\) small enough and depending on \(\alpha \). While assumption \((\widetilde{A})\) is still satisfied, assumption \((\widetilde{B}_0)\) is not satisfied anymore. Indeed, the curves involved in condition \((\widetilde{A})\) are holomorphic planes with simple asymptote in the fibers over minima of \(f\) (these have index \(0\)) and holomorphic planes with simple asymptote in the fibers over the unstable manifolds of critical points of \(f\) having index \(1\) (these have index \(1\)). Since these holomorphic planes are regular before perturbation, they remain regular for \(J_\epsilon \); thus \((\widetilde{A})\) is satisfied. To see that \((\widetilde{B}_0)\) is not satisfied, let us consider a pair of pants which doubly covers a trivial cylinder over a simple orbit in the fiber, capped at one puncture by a rigid plane in the same fiber. If the fiber lies over a critical point of \(f\) of index \(k\), the index of such a pair of pants is \(2k\), which is \(\le 0\) as soon as \(k\ge 2\). Such a curve always exists and cannot therefore be regular.
The paragraph that follows is conjectural: once the Morse–Bott techniques in [1] will be implemented along the lines of [3] within the context of linearized contact homology, they will provide a proof of Proposition 10. Then Proposition 11 is to be read as expressing the fact that a long exact sequence as the one in Theorem 1 is compatible with the computation in Proposition 10.
Remark 1.1
Nelson [8] has recently exhibited a large class of \(3\)dimensional contact manifolds for which transversality can be achieved for cylindrical contact homology.
3 The map \(D\)
It was pointed out to us by T. Ekholm that Proposition 8 is true as stated only in the absence of bad orbits. In case there are bad orbits, a term is missing in the definition of the chain map which induces the map \(D\). We describe now this additional term and complete the proof of Proposition 8.
Proposition 2.1
The map \(\Delta +\Delta ^{bad}\) defined by (85) and (2.1) is a chain map, and induces in homology the map \(D\) in the long exact sequence of Theorem 1.

\(\delta ^0\) vanishes on \(CM_*^{good}\oplus Cm_*^{good} \oplus CM_*^{bad}\) and \(\delta ^0:Cm_*^{bad}\rightarrow CM_{*1}^{bad}\) is an isomorphism over \({\mathbb {Q}}\).

\(\delta ^1({\textit{CM}}_*^{bad})=0\), \(\delta ^1({\textit{CM}}_*^{good})\subset {\textit{CM}}_{*1}\), \(\delta ^1({\textit{Cm}}_*)\subset {\textit{Cm}}_{*1}^{good}\).

\(\delta ^2({\textit{Cm}}_*)=0\), \(\delta ^2({\textit{CM}}_*)\subset {\textit{Cm}}_{*1}\) and \(\delta ^2({\textit{CM}}_*^{bad})\subset {\textit{Cm}}_{*1}^{good}\).
Remark 2.2
For a general filtered complex with differential \(\delta =\delta ^0+\delta ^1+\delta ^2+\dots \), the differential \(\bar{\delta }^2\) on the second page of the spectral sequence is not induced by a chain map on the first page. In our situation it was possible to exhibit a chain map inducing \(\bar{\delta }^2\) thanks to the fact that \(\delta ^0\) could be absorbed in an acyclic complex.
Notes
Acknowledgments
We would like to thank M. Abouzaid, T. Ekholm, J. Fish, J. Latschev, and J. Nelson for their comments on our original article. F. B.: Partially supported by ERC Starting Grant StG239781ContactMath. A. O.: Partially supported by ERC Starting Grant StG259118Stein. The present work is part of the authors activities within CAST, a Research Network Program of the European Science Foundation.
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