Suppose that \(\Omega^{2n}\) is a Liouville domain and consider a Legendrian submanifold \(\Lambda\subset\partial\Omega\). The primary goal of this paper is to bound the length of the shortest non-constant Reeb chord of \(\Lambda\) from above. Here the Reeb flow is associated to the canonical contact form on \(\partial\Omega\).

Let \(W\) denote the Liouville manifold obtained by completion of \(\Omega\). Our main results are phrased in terms of the symplectic cohomology \(\mathit{SH}(W)\), and its additional structures:

1. \(\mathit{PSS}:H^{*}(W)\to \mathit{SH}(W)\),

2. BV operator \(\Delta:\mathit{SH}(W)\to \mathit{SH}(W)\),

3. pair-of-pants product \(\ast:\mathit{SH}(W)\otimes \mathit{SH}(W)\to \mathit{SH}(W)\).

The symplectic cohomology \(\mathit{SH}(W)\) and its three additional structures are well-known; see, e.g., [1]–[8]. For further details on the conventions we use, see [9].

This leads us to our main definition:

Here we comment that \(n\)-invertible domains always have dimension \(2n\), i.e., we adopt the somewhat common convention in symplectic geometry that \(n\) stands for “half of the dimension.” The definition is closely related to that of string point invertibility introduced in [10].

Let us set some terminology before proceeding to our main theorem.

• The domain \(\Omega\) is completed to a Liouville manifold \(W\), which means that \(\Omega\subset W\) is embedded in such a way that the Liouville flow lines passing through the contact boundary \(\Omega\) define the symplectization end \(SY\) of \(W\) (here \(Y\simeq \partial \Omega\) is the ideal boundary).

• The radial function \(r\) is \(1\)-homogeneous \(\mathit{d}r(Z)=r\) and satisfies \(r|_{\partial \Omega}=1\). This function is smooth on the symplectization end \(SY\) and extends continuously to the rest of \(\Omega\).

• The Reeb vector field is defined to be \(R=X_{r}\) (defined on \(SY\)).

• The Floer cohomology group \(\mathit{SH}_{c}(\Omega)\) is defined as the “category theoretic limit” of Floer cohomologies of Hamiltonian systems whose generating vector field agree with \(cR\) outside of a compact set. We restrict \(c\) to be in the complement of the spectrum \(\mathit{Spec}(\Omega)\) of periods of \(R\). We assume throughout the body of the paper that Floer cohomologies are computed using coefficients in a field of characteristic two.

The spaces \(\mathit{SH}_{c}(\Omega)\), for \(c\) in the complement of the spectrum, form a persistence module (in the sense of [11]); the colimit of this persistence module is easily seen to be the symplectic cohomology \(\mathit{SH}(W)\).

The three structures \(\mathit{PSS},\Delta,\ast\) can be lifted to the persistence module as:

1. natural maps \(\mathit{PSS}:H^{*}(W)\to \mathit{SH}_{c}(\Omega)\) for \(c>0\),

2. natural endomorphisms \(\Delta:\mathit{SH}_{c}(\Omega)\to \mathit{SH}_{c}(\Omega)\),

3. natural products \(\ast:\mathit{SH}_{c_{1}}(\Omega)\otimes \mathit{SH}_{c_{2}}(\Omega)\to \mathit{SH}_{c_{1}+c_{2}}(\Omega)\).

We now attempt to “quantify” the notion of \(n\)-invertibility; first set: \[\mathscr{I}_{0,c}=\mathrm{span}\left\{ \mathit{PSS}(H^{n}(\Omega))\ast \mathit{SH}_{c'}(\Omega):c'<c\right\},\] where \(\mathit{PSS}(H^{n}(\Omega))\) takes values in \(\mathit{SH}_{c-c'}(\Omega)\). Similarly define for \(k\ge 1\): \[\mathscr{I}_{k,c}=\mathscr{I}_{k-1,c}+\mathrm{span}\left\{\Delta_{c_{1}}(\mathscr{I}_{k-1,c_{1}})\ast \mathit{SH}_{c_{2}}:c_{i}>0\text{ and }c_{1}+c_{2}=c\right\},\] and let \(\mathscr{I}_{c}\subset \mathit{SH}_{c}(\Omega)\) be the union of \(\mathscr{I}_{k,c}\) over all \(k\). Note that, since \(\mathit{SH}_{c}(\Omega)\) is a finite dimensional vector space, the colimit \(\mathscr{I}_{c}=\mathscr{I}_{c,k}\) is attained for some finite number \(k\). This subspace \(\mathscr{I}_{c}\subset \mathit{SH}_{c}(\Omega)\) is a natural subspace of the persistence module (i.e., continuation maps send \(\mathscr{I}_{c'}\) into \(\mathscr{I}_{c}\) for \(c'<c\)), and so the colimit of \(\mathscr{I}_{c}\) is a subspace of the symplectic cohomology \(\mathscr{I}\subset \mathit{SH}(W)\).

This leads to our second main definition in terms of the persistence refinement of the ideal \(\mathscr{I}\):

Our main result is:

0.0.1 Domains with vanishing symplectic cohomology↩︎

If \(\mathit{SH}(W)=0\), then the conclusion of Theorem 1 holds since the condition from Definition [definition:n-invertible] is trivially satisfied. This recovers a result of Zhou [7].

0.0.2 Cotangent bundles of products of spheres↩︎

This proves that \(T^{*}(S^{n_{1}}\times \dots \times S^{n_{k}})\) is \(n\)-invertible,¹ due to the morphism of BV-algebras established in [1], [2], [4], [9], [13]: \[H_{*}(\Lambda M)\to \mathit{SH}(T^{*}M)\] sending \([\mathit{pt}]\) to \(\mathit{PSS}(F)\) where \(F\) is the cohomology class represented by the cotangent fiber. Moreover, our argument also gives estimates on the \(n\)-invertibility capacity in terms of the lengths of the loops appearing in the classes \(A,B_{1},\dots,B_{k}\), where the length functional is the one determined by the domain \(\Omega\subset T^{*}M\) considered in [9]. As a consequence:

Let \(\Omega\) be a Liouville domain. Recently, there has been interest [7], [14] in bounding on the rationality constant of a non-exact Lagrangian inside \(\Omega\). Here we recall that a Lagrangian \(L\subset \Omega\) determines a Liouville class \([\lambda|_{L}]\in H^{1}_{\mathrm{dR}}(L)\), and the rationality constant \(\rho(L)\) is the infimal positive period of this class.

Our second main result is the following bound on the rationality constants of aspherical² Lagrangians inside of \(T^{*}T^{n}\):

It is noteworthy that our result applies to \(T^{*}T^{n}\), whereas previous results concerning aspherical Lagrangians \(L\) such as [7], [15] do not apply when the ambient space is \(T^{*}T^{n}\) (but rather apply when the ambient space behaves like the cotangent bundle of a non-aspherical manifold).

The first step used in the proof of Theorem 1 is a variation of the geometric construction similar to [16] and [17]. The result can be considered as a version of the \(1\)-jet neighborhood theorem for Legendrians.

It is clear that if \(\Lambda\) has no Reeb chords for the contact form \(\alpha\) (as in the statement of Theorem 4) of periods up to \(a\), then we can find an isotopy \(\Lambda_{t}\) satisfying the hypotheses of Theorem 4, and thereby conclude an exact embedding \(D_{\epsilon}T^{*}\Lambda\times D(a)\to \Omega\).

The next step is to appeal to the well-known Viterbo restriction (or Viterbo transfer) map, whose properties are summarized in the following theorem:

Thus, given a Legendrian \(\Lambda\) which has no chords of length up to \(a\), every smooth Liouville subdomain \(K\subset D_{\epsilon}T^{*}\Lambda \times D(a)\) receives a map: \[\mathfrak{R}:\mathit{SH}_{c}(\Omega)\to \mathit{SH}_{b}(K)\] compatible with structures, as in Theorem 5. In particular, if \(a\) is larger than \(c_{\mathrm{ni}}(\Omega)\), then we can pick \(c_{\mathrm{ni}}(\Omega)<c<b<a\) so that: \[1=0\in \mathit{SH}_{b}(K).\] This is because \(1\) lies in \(\mathscr{I}_{c}(\Omega)\), and \(\mathfrak{R}(\mathscr{I}_{c}(\Omega))\) vanishes in \(\mathit{SH}_{b}(K)\) since: \[\mathfrak{R}:H^{n}(\Omega)\to H^{n}(K)\] vanishes (as there are no non-trivial degree \(n\) classes in \(D_{\epsilon}T^{*}\Lambda\times D(a)\)).

The next step uses the following filtered Künneth map result:

Combining all these steps, we conclude that, if \(\Lambda\) admits no Reeb chords of length up to some number \(a>c_{\mathrm{ni}}(\Omega)\), then the unit vanishes in \(\mathit{SH}(T^{*}\Lambda)\). However, due to the comparison between string topology and symplectic cohomology, the unit cannot vanish in \(\mathit{SH}(T^{*}\Lambda)\). This provides the desired contradiction, and proves Theorem 1, (modulo Theorems 4, 5, and 6).

Turning now to Theorem 2, the method of proof is to use the framework of [9], which allows one to prove the desired relations in the BV algebra \(H_{*}(\Lambda M)\) by direct manipulation of smooth maps taking values in \(M\). The proof is entirely topological and all Floer theory is isolated in our earlier work [9].

Finally we discuss the proof of Theorem 3 concerning the rationality constants of aspherical Lagrangians \(L\) inside of \(T^{*}T^{n}\). The key idea is again to use a Viterbo restriction map; however, since the Lagrangian \(L\) is non-exact, Weinstein neighborhoods of \(L\) are not embedded into \(T^{*}T^{n}\) in an exact way, and so one cannot appeal directly to Theorem 5. One instead appeals to the so-called truncated Viterbo restriction map introduced by [7].

As in [7], given a Lagrangian \(L\subset \Omega\) we apply Theorem 7 to a small Weinstein neighborhood of \(L\). The next step in the argument is to apply the truncated Viterbo restriction map \(\mathfrak{R}\) to classes \(A_{i}^{\pm}\) as in Remark [remark:on-the-constant-C] to conclude classes in \(SH(T^{*}L,\kappa)\) (assuming the rationality constant of \(L\) is large enough). Using the fact that \(L\) is aspherical enables the use of two algebraic topology results [18], which we use to show \(\pi_{1}(L,\mathit{pt})\) admit a subgroup \(\mathbb{Z}^{n}\) whose associated covering space is open. This leads to a contradiction, since the covering space is also an aspherical \(n\)-manifold, and therefore must be homotopy equivalent to \(K(\mathbb{Z}^{n},1)\simeq T^{n}\) which has non-zero \(n\)th homology group (and so cannot be open). The details of this argument are given in §[sec:sec:proof-rationality].

The chord conjecture due to Arnol’d appears in [19], and it is formulated for \(S^3\) with the standard contact structure. Corollary 1 solves the Arnol’d chord conjecture for \(\Lambda\subset ST^{*}M\) when it applies.

0.0.3 Higher dilations↩︎

If one allows constraints on the topology of \(\Lambda\) relative the topology of \(M\), then one can deduce quite a bit by combining the following two propositions:

By appealing to [5], [7] concludes:

0.0.4 Orientation tricks↩︎

Let us note one curiously general example:

0.0.5 Aspherical ambient spaces↩︎

Let us comment that Corollary 1 applies to \(ST^{*}T^{n}\), because \(T^{n}\) is a Cartesian product of \(n\) copies of \(S^{1}\). This is notable because \(T^{n}\) is aspherical and hence the symplectic cohomology does not admit any type of higher dilation classes, or any trickery with orientations as in Claim [claim:spin-CPd]. In this respect, i.e., applicability to aspherical manifolds, our result is closer to the work of [28] which uses the product structure to bound the Hofer Zehnder capacity of cotangent bundles of manifolds admitting circle actions with non-contractible orbit class.

0.0.6 Fillable Legendrians↩︎

One can also restrict the class of Legendrians in another ways: by requiring that they admit Lagrangian fillings; see, e.g., [29]. If the fillings are assumed to be exact, then one can use wrapped Floer cohomology; see, e.g., [30]. The Arnol’d chord conjecture is established for conormal Legendrians via a comparison with the Morse homology of path spaces, see [31] which builds on work of [13], [32].

0.0.7 Strong Arnol’d chord conjecture↩︎

An interesting research direction is the “strong” version of the conjecture, as studied in [33], [34], which asks whether there is always a chord with distinct endpoints. It seems difficult to establish the strong Arnol’d conjecture from our methods, since the upper bound on the length of the Reeb chord is always at least the systole of \(\lambda|_{\partial\Omega}\) (so we cannot a priori guarantee chords shorter than all orbits, which is the surefire way to conclude the strong Arnol’d conjecture).

0.0.8 Time-dependent Reeb flows↩︎

The method is robust, in that it allows one to prove the existence of chords for time-dependent Reeb flows, as long as they are uniformly fast. This is clear from the statement of Theorem 4. We note that other methods of producing Reeb chords (such as bubbling analysis for almost complex structures of SFT type) do not seem to easily produce chords for time-dependent flows.

0.0.9 Cotangent bundles of manifolds with negative sectional curvature↩︎

The restrictions from this paper are related to interactions between product structures, BV operators, and the PSS morphisms. When \(\Omega\subset T^{*}M\), and \(M\) has negative sectional curvature, there are seemingly not enough relations between the top degree PSS classes and the other classes to apply our methods. It is natural to wonder if there a symplectic cohomology based upper bound on the length of Reeb chords of Legendrian knots in \(\partial\Omega\) for domains \(\Omega\subset T^{*}\Sigma_{g}\), where \(\Sigma_{g}\) is a surface of genus \(g>1\). Note that, since \(ST^{*}\Sigma_{g}\) is a contact 3-manifold, the existence of Reeb chords is known [35], [36]. One can also compare with the problem on the finiteness of the Hofer-Zehnder capacity of compact subsets in \(T^{*}\Sigma_{g}\), [37], [38].

The authors would like to thank E. Shelukhin for many useful discussions surrounding these topics. The key ideas for this project were developed during the second author’s visit to Augsburg in May 2025; the authors would like to thank Universität Augsburg for providing a pleasant work environment. The second author thanks P. Biran at ETHZ, J. Shang at LMO, J. Zhang at USTC IGP, W. Gong at BNU, and P. Salomão at SUSTech SICM for valuable discussions during the preparation of this work. The authors also wish to warmly thank Z. Zhou and K. Guo for communicating their related work [39] (developed contemporaneously with work the present paper) during the May 2026 program at the Tianyuan Mathematics Research Center in Kunming.

The first author was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – 541525489. The second author was funded by the ANR project CoSy and an NSERC/CRSNG Discovery grant.

The contents of this section are:

• §[sec:sec:from-legendr-stab] on the existence of embeddings of stabilized codisk bundles;

• §[sec:sec:kunneth-morphism] on the Künneth morphism;

• §[sec:sec:string-topology-products-spheres] on the string topology of products of spheres;

• §[sec:sec:proof-rationality] on rationality constant of Lagrangians in \(T^{*}T^{n}\).

Due to the more-or-less standard nature of the Viterbo restriction map, see [1], [29], [30], [40], and the fact the truncated version already appears in [7], we skip directly from §[sec:sec:from-legendr-stab] to §[sec:sec:kunneth-morphism] and confine the proof of Theorems 5 and 7 to Appendix [sec:sec:viterbo-restriction].

The goal of this section is to prove Theorem 4. The first part of the construction involves a coisotropic embedding:

The next step is to appeal to the following general fact:

We apply this fact with: \[N=\Lambda\times D(a)\quad M_{1}=T^{*}\Lambda\times \mathbb{C}\quad M_{2}=\Omega.\] In particular, we conclude from Lemma 3 a symplectic embedding: \[\Phi:D_{\epsilon}T^{*}\Lambda\times D(a)\to \Omega\] which agrees with the map \(\varphi\) from Lemma 2 on \(\Lambda\times D(a)\). As it is a symplectic embedding, it follows that: \[\Phi^{*}\lambda=p\mathit{d}q+\lambda_{D(a)}+\beta\] where \(\beta\) is a closed 1-form (here \(p\mathit{d}q\) is the standard Liouville form on \(T^{*}\Lambda\)). Since \(\Phi\) agrees with \(\varphi\) on the zero section, and \(D_{\epsilon}T^{*}\Lambda\times D(a)\) deformation retracts onto \(\Lambda\times D(a)\) we conclude that all periods of \(\beta\) vanish, and hence \(\beta\) is an exact 1-form. This completes the proof of Theorem 4. \(\square\)

A Liouville manifold \((W,\omega,Z)\) involves the data of a Liouville vector field \(Z\) such that \(L_{Z}\omega=\omega\). Let us denote by \(Z_{s}\) the time-\(s\) flow. Hamiltonian Floer theory of \((W,\omega,Z)\) is well-established provided one uses Hamiltonian systems and \(\omega\)-tame almost complex structures that are invariant under \(Z_{s}\), \(s\ge 0\), outside of a compact set; see, e.g., [43]. It is in this context that one defines the persistence module \(\mathit{SH}_{c}(\Omega)\) used throughout this paper.

The product \(W\times \mathbb{C}\) is another Liouville manifold with diagonal Liouville vector field: \[Z_{W\times \mathbb{C}}=Z_{W}+Z_{\mathbb{C}}\] where \(Z_{\mathbb{C}}\) is the radial Liouville vector field on \(\mathbb{C}\), with its standard symplectic form.

The naive idea of the Künneth morphism is that, if \(H_{t}\) generates a non-degenerate Hamiltonian system on \(W\) (with positive slope \(b\) as in the statement of Theorem 6), then: \[H_{t}+\tau \pi\left|z\right|^{2}\text{ where }\tau=b/a\] generates a non-degenerate Hamiltonian system on \(W\times \mathbb{C}\), and the vector space underlying the Floer complex \(\mathit{CF}(H_{t}+\tau\pi\left|z\right|^{2})\) is identified with the Floer complexes \(\mathit{CF}(H_{t})\) computed in \(W\). If one picks a split almost complex structure \(J_{W\times \mathbb{C}}=J_{W}\oplus J_{\mathbb{C}}\), then the differential respects this identification, and one obtains an isomorphism of complexes: \[\label{eq:isomorphism-of-complexes} \mathit{CF}(H_{t}+\tau \pi\left|z\right|^{2},J_{W\times \mathbb{C}})\to \mathit{CF}(H_{t},J_{W}).\tag{1}\] Since \(\tau\in (0,1)\), this chain level identification also respects the unit element (this is no longer true when \(\tau\not\in (0,1)\)).

The problem in using this observation in symplectic cohomology theory is that \(H_{t}+\pi \left|z\right|^{2}\) and \(J_{W}+J_{\mathbb{C}}\) are not \(Z\)-invariant outside of a compact set, and so our preferred framework (as used in [9], [43] and throughout the rest of this paper) cannot be applied. Existing literature [40], [44] successfully establishes a “Künneth isomorphism” between the colimits: \[\mathit{SH}(W\times \mathbb{C})\simeq \mathit{SH}(W)\otimes \mathit{SH}(\mathbb{C}).\] Such a statement is not ideal for our purposes since \(\mathit{SH}(\mathbb{C})=0\); we instead require a “filtered” Künneth map \(\mathfrak{K}\) as in Theorem 6.

0.0.10 Choice of almost complex structure↩︎

As in Theorem 6, fix:

• a Liouville domain \(Q\), with radial coordinate \(r_{1}\),

• an \(\omega\)-tame almost complex structure \(J_{1}\) on the completion \(W\), which is invariant under \(Z_{s}\), \(s\ge 0\), outside of \(Q=\left\{r_{1}\le 1\right\}\),

• the standard complex structure \(J_{2}\) on \(\mathbb{C}\),

• the radial coordinate \(r_{2}=a^{-1}\pi\left|z\right|^{2}\) for the Liouville domain \(D(a)\).

The complex structure: \[Z_{s}^{*}J_{1}=\mathit{d}Z_{s}^{-1}J_{1}\mathit{d}Z_{s}\] on \(W\) can be considered as “neck stretched,” loosely speaking.

Let \(\ell(r)\) be a smooth approximation of \(\max\left\{\log(r),0\right\}\), such that:

• \(\ell(r)=\log(r)\) when \(r\ge 1\),

• \(\ell(r)\le 0\) for \(r\le 1\),

• \(\ell(r)\) is constant on the interval \(r\in [0,\frac{1}{2}]\).

See §0.0.11 for an explicit choice of such \(\ell\).

Define an almost complex structure on \(T(W\times \mathbb{C})\): \[\label{eq:special-J} J=Z_{-\ell(r_{2})}^{*}(J_{1})\oplus J_{2}.\tag{2}\] Then we have:

0.0.11 Hamiltonian systems for the Künneth morphism↩︎

We continue with the set-up of §0.0.10. The construction of the Künneth map is defined in a similar way to the “naive” Künneth map described in §[sec:sec:kunneth-morphism]; see equation 1 . As in the statement of Theorem 6, we will find a domain \(K\subset Q\times D(a)\) and directly identify a chain complex computing \(\mathit{SH}_{b}(K)\) with a chain complex computing \(\mathit{SH}_{b}(Q)\); this identification produces the desired map \(\mathfrak{K}\).

The primary goals of §[sec:sec:kunneth-morphism] are to construct this map and prove it respects \(\mathit{PSS}(1)\); a secondary goal is to make the map \(\mathfrak{K}\) canonical. For the sake of canonicity, we introduce some functions via explicit formulas:

The map \(\mathfrak{K}\) is defined via the following family of Hamiltonian systems.

0.0.12 Floer cylinders for the Künneth morphism↩︎

In this section we prove structural results for the Floer complex \(\mathit{CF}(H_{t},J)\) provided \(H_{t}\in \mathscr{O}_{b}(\delta,\mathscr{U})\) and \(\delta\) and the open set \(\mathscr{U}\) are sufficiently small.

Due to Lemma 6, we can pick \(\delta>0\) and \(\mathscr{U}\) small enough so that, for each \(H_{t}\in \mathscr{O}_{b}(\delta,\mathscr{U})\), there is a tautological isomorphism of complexes: \[\mathfrak{K}_{H_{t}}:\mathit{CF}(H_{t},J)\simeq \mathit{CF}(H_{t}|_{W\times \left\{0\right\}},J|_{W\times \left\{0\right\}}).\] This map should be thought of as one representation of the Künneth map satisfying Theorem 6; in the next section we explain how to make \(\mathfrak{K}\) “canonical,” i.e., independent of the auxiliary choices used in \(H_{t}\).

0.0.13 Canonicity of the Künneth map↩︎

Recall from §[sec:sec:introduction] that \(\mathit{SH}_{b}(Q)\) is defined to be the (formal) inverse limit of the Floer cohomologies of systems which agree with \(b r_{Q}\) outside of a compact set. Thus, for \(\delta>0\) and \(\mathscr{U}\) small enough such that \(H_{t}\in \mathscr{O}_{b}(\delta,\mathscr{U})\) satisfies Lemma 6 we have the following diagram: \[\label{eq:definition-of-K} \begin{tikzcd} \mathit{SH}_{b}(K_{\delta})\arrow[d]\arrow[r,dashed,"\mathfrak{K}_{\delta}"]&\mathit{SH}_{b}(Q)\arrow[d]\\ \mathit{HF}(H_{t},J)\arrow[r,"\mathfrak{K}_{H_{t}}"]&\mathit{HF}(H_{t}|_{W\times \left\{0\right\}},J|_{W\times \left\{0\right\}}) \end{tikzcd}\tag{4}\] where \(K_{\delta}\subset Q\times D(a)\) is as in Definition 3. The upper horizontal map is well-defined, for fixed \(H_{t}\), by inverting the right vertical map. However, it a priori depends on the choice of \(P_{t}\) used in the definition of \(H_{t}\). In this section we show the result is independent of \(P_{t}\).

0.0.14 Compatibility of the Künneth map with the unit element↩︎

Fix \(\delta\) and \(\mathscr{U}\) small enough that Lemma 7 applies; this fixes a domain \(K_{\delta}\) and a map \(\mathfrak{K}_{\delta}:\mathit{SH}_{b}(K_{\delta})\to \mathit{SH}_{b}(Q)\). The goal of this section is proving:

In this section we prove Theorem 2 concerning classes in \(H_{*}(\Lambda M)\) for \(M=S^{n_{1}}\times \dots \times S^{n_{k}}\). First, we construct classes \(A\) and \(B\) for the case of \(M = S^n\). The key part of this step is the notion of completing manifolds as in [45] which relates classes in the spherical tangent bundle \(STS^{n}\) with classes in \(\Lambda S^{n}\). This step is done in §0.0.15. In §0.0.16 we construct a special class in \(H_{n}(STS^{n})\) using the notion of a generalized section. Afterwards in §0.0.18 we use the case of one sphere to construct the classes in \(M=S^{n_{1}}\times \dots \times S^{n_{k}}\).

Throughout this section we use the bordism class approach to the homology of the loop space, as in [9].

0.0.15 Completing manifolds↩︎

Let \(STS^n\) denote the spherical tangent bundle. Each point \(v\) determines its orthogonal equator, which is the unique equator passing through the basepoint of \(v\) and which is orthogonal to \(v\) at that basepoint. Consider the sphere bundle of pairs: \[E=\left\{(x,v)\in S^{n}\times STS^{n}:x\text{ lies in the orthogonal equator of }v\right\}.\] Define the map \(F: E \to \Lambda S^n\) which sends \((x, v) \in E\) to the circle obtained by intersecting \(S^n\) with the affine plane passing through \(x\) and \(v\) (this means the affine plane intersects the basepoint of \(v\) and is tangent to \(v\)). This circle is parametrized by \(\mathbb{R}/\mathbb{Z}\) with constant speed in the unique way that the tangent vector at \(0\) points in the direction of \(v\).

This procedure yields the completing manifolds operation: \[\gamma:H_*(STS^n) \to H_{*+n-1} (\Lambda S^n)\] as follows: given a class \(f: X \to STS^n\), the class \(\gamma f\) is the composition of the two upper horizontal morphisms in the following diagram: \[\begin{tikzcd} f^* E \arrow[r]\arrow[d] & E\arrow[d, "\pi"] \arrow[r] & \Lambda S^n \\ X \arrow[r, "f"] & STS^n. \end{tikzcd}\]

0.0.16 Generalized section↩︎

In this section we construct a special class: \[s: \mathbb{R}P^{n} \to STS^{n}\] to which we will apply the completing manifolds construction of §0.0.15 to obtain a class in \(H_{2n-1}(\Lambda S^{n})\). The construction of \(s\) is as a generalized section, i.e., that \(\pi \circ s : \mathbb{R}P^n \to S^n\) has degree \(1\). Even though \(STS^{n}\to S^{n}\) does not admit a section for even values of \(n\), it does admit a generalized section: after a blow up of the base space, transforming \(S^{n}\) to \(\mathbb{R}P^{n}\), it admits a section. We describe this with greater detail.

The first step is to recall the unoriented blow-down map \(b:\mathbb{R}P^{n}\to S^{n}\).

0.0.17 Manipulation of classes in the free loop space of a sphere↩︎

Apply the completing manifolds operation from §0.0.15 to the map \(s: \mathbb{R}P^n \to STS^n\) constructed in §0.0.16 to produce the class \(B=\gamma s\in H_{2n-1}(\Lambda S^{n})\). Similarly, let \(P\in H_{n-1}(\Lambda S^{n})\) be the class obtained from the point class in \(STS^{n}\), and let \(\mathit{pt}\) denote the point class in \(H_{0}(\Lambda S^{n})\) consisting of a single constant loop. Our first lemma in this section is:

Following the terminology in [9] introduce the action class: \[A(x;t)=(\cos(2\pi t)x_{0}-\sin(2\pi t)x_{1},\sin(2\pi t)x_{0}+\cos(2\pi t)x_{1},x_{2},\dots,x_{n}),\] associated to the rotation circle action so that \(A\in H_{n}(\Lambda S^{n})\), and similarly let \(A^{-1}(x;t)=A(x;-t)\). We have:

This concludes the first part of Theorem 2, since \(A\ast \Delta(B\ast \mathit{pt})=[S^{n}]\) by combining Lemmas 11, 12, and 13.

0.0.18 The product of spheres↩︎

For the second part on the product of spheres, we require the following notion:

In this section we prove Theorem 3. Fix a fiberwise starshaped subdomain \(\Omega\subset T^{*}L\), consider a compact Lagrangian submanifold \(L\) in the interior of \(\Omega\). The goal is to prove that either \(L\) is exact, or \(\rho(L)\) is bounded above by a constant which depends only on \(\Omega\).

0.0.19 Non-exact Weinstein neighbourhood↩︎

The first step is to use the Weinstein neighborhood theorem to conclude a symplectic embedding \(\iota:K\to \Omega\) of a disk cotangent bundle \(K=DT^{*}L\) which extends the embedding \(L\to \Omega\). It follows that the period group of the closed one-form: \[\iota^{*}\lambda_{\Omega}-\lambda_{K}\] is valued in \(\rho(L) \mathbb{Z}\). This holds because \(K\) deformation retracts onto \(L\) and so the periods of \(\iota^{*}\lambda_{\Omega}-\lambda_{K}\) are the same as the periods of \(\lambda_{\Omega}|_{L}\).

0.0.20 Classes in the free-loop space of cotangent bundle of a torus↩︎

The next step introduces certain classes in \(\mathit{SH}(T^{*}T^{n})\); these classes were alluded to in Remark [remark:on-the-constant-C].

Therefore, having fixed \(\Omega\subset T^{*}T^{n}\), we can pick the slope \(c>0\) large enough that \(A_{i}^{+}\ast A_{i}^{-}=\mathit{PSS}(1)\) holds in \(\mathit{SH}_{c}(\Omega)\) for each \(i=1,\dots,n\).

0.0.21 Truncated Viterbo restriction↩︎

Combining the set-up of §0.0.19 and §0.0.20, we can apply Theorem 7 to conclude:

0.0.22 From symplectic cohomology back to string topology↩︎

In this subsection, we import one of the main results from [4], [13] that there is a BV algebra map from \(\mathit{SH}(T^{*}L)\) to \(\mathit{HM}_{*}(\Lambda L)\). We emphasize here that \(\mathit{HM}_{*}(\Lambda L)\) uses a Morse model rather than the bordism classes model used in §[sec:sec:string-topology-products-spheres].

Referring to the notation and conclusions of Corollary 2, we conclude:

0.0.23 Two facts about aspherical manifolds↩︎

We recall two facts about the string topology aspherical manifolds taken directly from [18].

We apply these lemmas in our sitation. Let \(C_{i},\kappa_{i}\) be as in Corollary 3. Pick elements \(\gamma_{i}\in \kappa_{i}\) based at a chosen basepoint \(\mathit{pt}\in L\) so that we may consider \([\gamma_{i}]\in \pi_{1}(L,\mathit{pt})\).

0.0.24 Proof of Theorem 3↩︎

By Lemma 19, if \(\iota:L\to \Omega\) is a Lagrangian embedding with \(\rho(L)>2c\) where \(c\) is as in §0.0.20, then the kernel of \(\iota^{*}\lambda_{\Omega}\) contains a finite index subgroup \(G\). The kernel of a non-zero homomorphism to \(\mathbb{R}\) has an infinite index. This implies \(\iota^{*}\lambda_{\Omega}=0\), as desired.\(\square\)

The Viterbo restriction map, as discussed in [1], [7], [29], [30], [39], [40], is a map \(\mathfrak{R}:\mathit{SH}(\Omega)\to \mathit{SH} (K)\) associated to an exact embedding \(\iota:K\to \Omega\) of Liouville domains. This map can be refined to a filtered version, and this refinement respects additional structures; the precise statement we require in the body of the text is given in Theorem 5. The primary goal of this section is proving this result. Since it is quite well-understood, we have opted to place this part of the paper in an appendix.

As developed in [7] (and also used in [39]), there is also a “truncated Viterbo restriction map” for non-exact embeddings. We required a version of this in the proof of Theorem 3; the precise statement we used was given in Theorem 7 and we briefly review the proof in §[sec:sec:truncated-viterbo-restriction]

We recall the geometric set-up for Theorem 5: we let \(\iota:K\to \Omega\) be an exact embedding of Liouville domains, and suppose that the image of \(K\) is contained in the interior of \(\Omega\). To be precise, this means that \(K\) is a Liouville domain with its own Liouville form \(\lambda_{K}\), and the difference \(\lambda_{K}-\iota^{*}\lambda\) is an exact form. When no confusion will arise, we write \(K\subset \Omega\) and consider it as a subset of \(\Omega\).

As will be made clear in the sequel, a key ingredient in (our approach to) the Viterbo restriction map is the notion of: Floer cohomology of Hamiltonian in an action window. The case of relevance to us is the following: given a smooth, autonomous, Hamiltonian \(H\), and an action window \(I\) whose endpoints are not contained in the spectrum of action values of 1-periodic orbits of \(H\), there is a canonically defined vector space \(\mathit{HF}_{I}(H)\). We develop the necessary aspects of this theory in §[sec:sec:appendix-filtered].

The table of contents of this section is as follows: §[sec:sec:viterbo-restriction-system] discusses the special Hamiltonian system so that the Viterbo restriction map is represented as the relationship between a Floer complex (computing \(\mathit{SH}_{c}(\Omega)\)) and its quotient complex in a certain action window; §[sec:sec:no-escape-lemma] discusses the well-known “no-escape” lemma which shows that the aforementioned quotient complex is isomorphic to \(\mathit{SH}_{b}(K)\), and there we conclude the proof of Theorem 5. In §[sec:sec:truncated-viterbo-restriction] we discuss Zhou’s truncated version (Theorem 7).

Denote by \(r\) the radial coordinate for \(\Omega\) and by \(\rho\) the radial coordinate for \(K\). For \(b>c\), such that \(b\not\in \mathit{Spec}(K)\) and \(c\not\in \mathit{Spec}(\Omega)\), consider the non-smooth Hamiltonian: \[\label{eq:VR-system-non-smooth} H_{a,c}=\left\{ \begin{align} &b\rho&&\text{ in }K\\ &b&&\text{ in }\Omega-K\\ &cr+b-c&&\text{ in }W-\Omega, \end{align} \right.\tag{12}\] where \(W\) denotes the completion of \(\Omega\). Because this system is not smooth, it does not have an obviously defined Hamiltonian Floer theory. Thus we first digress on how to discuss the Floer cohomology of \(H_{b,c}\).

0.0.25 Mollification formula↩︎

For each number \(\delta\) such that \(K\subset \left\{r<\delta\right\}\), consider the function (see Figure 2): \[\label{eq:non-smooth-first-pass} H_{\delta,b,c}= \left\{ \begin{align} &2\delta b&&\text{ on }\rho\le \delta,\\ &b\rho+\delta b&&\text{ on }\rho\in [\delta,1-\delta],\\ &b&&\text{ on }\rho\in [1-\delta,1]\text{ and in }\Omega-K,\\ &cr+b-c&&\text{ on }r\ge 1. \end{align} \right.\tag{13}\]

We will smooth this function using convolution with a mollifier.

Associated to such \(f\), we denote the convolution by the formula: \[(fH)_{\delta,b,c}=f_{\delta}\ast H_{\delta,b,c}\] where \(f_{\delta}(x)=\delta^{-1}f(\delta x)\). It is defined by the usual convolution formula from real analysis, except that one uses either \(\rho\) or \(r\) depending on the region of the domain in 13 . Because the mollification only sees parts of the domain within values \(\pm \delta\) for the coordinates \(\rho\) and \(r\), it is readily checked that the function is smooth on all of \(\Omega\).

As \((fH)_{\delta,b,c}\) is a smooth function, it has a well-defined filtered Floer cohomologies \(\mathit{HF}_{I}((fH)_{\delta,b,c})\) for every action window \(I\subset \mathbb{R}\) that is disjoint from the spectrum of action values of \((fH)_{\delta,b,c}\), as explained in §[sec:sec:appendix-filtered].

0.0.26 The action spectrum of the Viterbo restriction system↩︎

In this section we prove a result involving certain action windows for the smoothed Hamiltonian systems \((fH)_{\delta,b,c}\).

0.0.27 Canonical subquotient map↩︎

Let us denote by \(\mathscr{T}(b,c)\) the category of tuples \(\theta=(\delta,f)\) so that \((b,c,\delta,f)\) are admissible in the sense of Definition 6. By the result of Lemma 20, for each \(\theta\in \mathscr{T}(b,c)\) one has a quotient map: \[\label{eq:canonical-subquotient-map} \mathfrak{Q}_{\theta}:\mathit{HF}((fH)_{\delta,b,c})\to \mathit{HF}_{I}((fH)_{\delta,b,c})\text{ where I(\theta)=[0,\tau], \tau=3b\delta.}\tag{14}\] Technically we have two maps: \[\mathit{HF}\to \mathit{HF}_{(-\infty,3b\delta]}\leftarrow \mathit{HF}_{[0,3b\delta]},\] but because there are no orbits with action in \((-\infty,0]\) the second arrow is an isomorphism and can be inverted; see Remark [remark:enlarging-intervals].

For future notational convenience, let us denote the domain and codomain of \(\mathfrak{Q}_{\theta}\) by \(T_{\theta}\) and \(Q_{\theta}\) (for “total” and “quotient,” respectively).

The goal in this section is to explain why this map is independent of the choice of auxiliary data \(\delta,f\), in the sense that there is a canonical functor: \[\theta\in \mathscr{T}(b,c)\to \mathfrak{Q}_{\theta}\in \text{(category of linear maps)}.\] when \(\mathscr{T}(b,c)\) is considered as an indiscrete groupoid (i.e., a category with a unique morphism between any two objects). What this entails is constructing for each \(\theta,\theta'\) maps \(\mathfrak{c}_{\theta,\theta'}\) such that:

1. the diagram commutes: \[\begin{tikzcd} T_{\theta}\arrow[r,"\mathfrak{c}_{\theta,\theta'}"]\arrow[d,"\mathfrak{Q}_{\theta}",swap]& T_{\theta'}\arrow[d,"\mathfrak{Q}_{\theta'}"]\\ Q_{\theta}\arrow[r,"\mathfrak{c}_{\theta,\theta'}"]&Q_{\theta'} \end{tikzcd}\]

2. \(c_{\theta,\theta'}=\mathit{id}\) if \(\theta=\theta'\),

3. for all triples \(\theta,\theta',\theta''\) one has \(\mathfrak{c}_{\theta',\theta''}\circ \mathfrak{c}_{\theta,\theta'}\).

For related discussion, see [46].

The construction of the maps \(\mathfrak{c}_{\theta,\theta'}\) uses the simplest type of continuation map, namely those associated to the linear interpolation between \((fH)_{\delta,b,c}\) and \((f'H)_{\delta',b,c}\), as in §[sec:sec:floer-cohom-smooth].

Let us agree to also denote by \(\mathfrak{Q}_{b,c}:\mathit{SH}_{c}(\Omega)\to \mathit{Q}_{b,c}\) the resulting map.

0.0.28 Compatibility with additional structures↩︎

In this section we use the results of §[sec:sec:comp-struct-1] to deduce that the maps \(\mathfrak{Q}_{b,c}\) constructed in the previous section respect additional structures.

The easy statements to conclude (using §[sec:sec:comp-struct-1]) are:

• \(\mathfrak{Q}_{b,c}\) commutes with \(\Delta\),

• \(\mathfrak{Q}_{b,c}\) commutes with \(\mathit{PSS}\);

the statements involving the product and the continuation map are a bit more involved as they involving changing the slopes \(b,c\).

We use the no-escape lemma [30]. This will be used to ensure that solutions to the Floer equations with all asymptotics in \(K\subset \Omega\) remain localized to \(K\). The algebraic consequence of this lemma is that the group \(Q_{b,c}\) in §0.0.27 is canonically isomorphic to \(\mathit{SH}_{b}(K)\). Composing this with the previously defined map \(\mathfrak{Q}:\mathit{SH}_{c}(\Omega)\to Q_{b,c}\) yields a map \(\mathfrak{R}:\mathit{SH}_{c}(\Omega)\to \mathit{SH}_{b}(K)\) which is the Viterbo restriction map satisfying Theorem 5.

In the previous section §[sec:sec:viterbo-restriction-system], we were able to be rather agnostic about the choice of almost structure \(J\); we only required that was \(\omega\)-tame and \(Z\) invariant outside of some large compact set. In this section we will need to impose a futher condition defined in terms of the radial coordinate \(\rho\) for \(K\):

• \(J\) satisfies \(\lambda_{K}\circ J=\mathit{d}\rho\) near the level set \(\rho=1/2\).

In [7], a statement similar to Theorem 7 is proven; namely, that there is a Viterbo restriction map \(\mathit{SH}_{b}(\Omega)\to \mathit{SH}(T^{*}L)\) provided that that the rationality constant \(\rho\) of a Lagrangian embedding \(L\subset \Omega\) is larger than \(2b\).

The overall structure of the proof of Theorem 7 is exceeding similar to the proof of Theorem 5; the main difference occurs in §0.0.26. The argument introduced orbits of types [item:T1], [item:T2], and [item:T3] (see Figure 2). The non-exactness of the embedding implies that action values of orbits of types [item:T1] and [item:T2] computed in \(\Omega\) differs from the action values computed in \(K\) by addition of the of \(\beta=\lambda_{K}-\iota^{*}\lambda_{\Omega}\) over the orbit. To be precise:

• orbits of type [item:T1] have action in \((b\delta,2b\delta]+\rho\mathbb{Z}\),

• orbits of type [item:T2] have action in \((b-\sigma(b),b]+\rho\mathbb{Z}\),

• orbits of type [item:T3] have action in \((b-c,b]\), as before.

The key ingredient for the proof to proceed as in the exact case is:

The rest of the proof proceeds as in the exact case. We comment only that the conclusion that all orbits in the action window \([0,3b\delta]\) have \(\int \beta=0\) is used again to obtain the no-escape lemma, and to obtain the conclusion on the free homotopy classes given in Theorem 7.

In this appendix, we review the theory of the filtered Floer cohomology groups associated to a smooth autonomous Hamiltonian \(H\). Throughout we focus on the case relevant to the body of the paper:

We also fix an \(\omega\)-tame almost complex structure \(J\) that is Liouville invariant outside of a compact set (and will refer to such almost complex structures as admissible). To simplify the exposition, as in Remark [remark:significant-canonical], we fix \(J\) rather than treat it as an auxiliary choice.

Any admissible Hamiltonian \(H\) as above has a well-defined spectrum of action values: \[\mathit{Spec}(H):=\left\{\int_{0}^{1} H(x(t))dt-x^{*}\lambda:x\text{ is a 1-periodic orbit of }X_{H}\right\}.\] This is a closed and nowhere dense subset of the real line.

Let us denote by \(\mathscr{O}(\epsilon)\) the indiscrete groupoid whose objects are choices of \(\epsilon\)-admissible perturbation, that is to say, there is a unique morphism between any two objects.

Recall that a Hamiltonian connection is a convenient way to package the inhomogeneous term in Floer’s equation; for a more detailed introduction introduction, we refer the reader to [49] and [9]. Hamiltonian connections over a surface \(\Sigma\) are a special type of Ehresmann connection on the fibre bundle \(\Sigma\times W\to \Sigma\).

Hamiltonian connections \(\mathfrak{H}\) are defined by connection potentials, namely \(1\)-forms \(\mathfrak{a}\in \Omega^{1}(\Sigma\times W)\) which vanish on the vertical distribution (tangent to the fibres \(\left\{\mathit{pt}\right\}\times W\)). In local coordinates \(s+it\) on \(\Sigma\), such a potential can be expressed as: \[\mathfrak{a}=K_{s,t}\mathit{d}s+H_{s,t}\mathit{d}t\] where \(K_{s,t},H_{s,t}\) are functions on \(W\times \mathbb{C}\). The connection \(\mathfrak{H}\) is defined to be the “orthogonal complement” of the vertical distribution with respect to the two-form \(\omega-\mathit{d}\mathfrak{a}\). More prosaically, \(\mathfrak{H}\) is locally spanned by the vector fields: \[\partial_{s}+X_{K_{s,t}}\quad\text{and}\quad\partial_{t}+X_{H_{s,t}}.\] This concludes are recollection of Hamiltonian connections.

Let \(\Sigma\) be a Riemann surface with \(N+1\) cylindrical ends, decomposed into \(N\) inputs and \(1\) output and let \(\mathfrak{H}\) be a Hamiltonian connection over \(\Sigma\) whose connection potential \(\mathfrak{a}\) is locally expressed as \(\mathfrak{a}=K_{s,t}\mathit{d}s+H_{s,t}\mathit{d}t\) where \(K_{s,t},H_{s,t}\) are admissible for all \(s,t\) in the sense of Definition 7; in the cylindrical ends we suppose \(\mathfrak{a}\) is written as: \[\mathfrak{a}=H_{k}dt\text{ for }k=1,\dots,N,\infty,\] where \(\infty\) corresponds to the unique output. We assume that all \(H_{k}\) are admissible non-discriminant Hamiltonians as in §[sec:sec:floer-cohom-smooth].

Following the well-known recipe as in [49] and, say, [9], [50], for each admissible almost complex structure \(J\) and generic perturbation \(\mathfrak{p}\) of \(\mathfrak{H}\), one obtains a linear map: \[\label{eq:general-tqft} \mathit{CF}(H_{1}+D_{1,t},J)\otimes\dots\otimes \mathit{CF}(H_{N}+D_{N,t},J)\to \mathit{CF}(H_{\infty}+D_{\infty,t},J).\tag{16}\] Here the perturbation term \(\mathfrak{p}\) of \(\mathfrak{H}\) carries with it perturbations \(D_{k,t}\in \mathscr{O}(\epsilon)\), for \(k=1,\dots,N,\infty\). A variant of the construction considers smooth families of domains \(\Sigma_{\tau}\) and connections \(\mathfrak{H}_{\tau}\) and counts the rigid elements appearing in a parametric moduli space (this is how chain homotopy terms and the BV operator are defined).

We wish to understand how such operations interact with action windows. The change in action values is governed by the so-called “curvature” term associated to the Hamiltonian connection.

In the next subsections §0.0.29, §0.0.30, §0.0.31 we will analyze in more detail the operations needed in the body of the paper.

0.0.29 The pair of pants product↩︎

The interaction between action windows and the product structure is a bit subtle. The reason is that defining a binary operation with action windows \(I_{1},I_{2},I_{\infty}\), say: \[\label{eq:product-map-windows} \mu:\mathit{HF}_{I_{1}}(H_{1})\otimes \mathit{HF}_{I_{2}}(H_{2})\to \mathit{HF}_{I_{\infty}}(H_{\infty}),\tag{17}\] typically requires that the ideal subcomplex: \[\label{eq:subcomplex-ideal} \mathit{CF}_{A>\sup I_{1}}\otimes \mathit{CF}_{A>\inf I_{2}} + \mathit{CF}_{A>\inf I_{1}}\otimes \mathit{CF}_{A>\sup I_{2}}\tag{18}\] is mapped to zero, rather than just the subcomplex \(\mathit{CF}_{A>\sup I_{1}}\otimes \mathit{CF}_{A>\sup I_{2}}\).

Fix \(\mathfrak{H}\) on the pair of pants surface \(\Sigma=\mathbb{C}P^{1}-\left\{1,2,\infty\right\}\) with standard cylindrical ends as in §[sec:sec:interf-with-tqft].

We will now explain this with more details and give a proper proof keeping track of the perturbation terms \(\mathfrak{p}\) and using the formal definition of the invariant.

To conclude this section, we explain how Lemma 30 induces a canonical operation 17 when the hypotheses of Lemma 29 are satisfied.

0.0.30 Continuation maps↩︎

A continuation map depends on a Hamiltonian connection \(\mathfrak{H}\) on a cylinder. The resulting operation is a map: \[\label{eq:continuation-map} \mathfrak{c}_{\mathfrak{H}}:\mathit{HF}_{I_{0}}(H_{0})\to \mathit{HF}_{I_{1}}(H_{1})\tag{19}\] for suitable action windows \(I_{i}\), \(i=0,1\). The main structural theorem (and analogue of Lemma 29) is:

0.0.31 BV operator on filtered Floer cohomology↩︎

Let \(H\) be an admissible and non-discriminant Hamiltonian \(H\) as in Definition 7 and let \(I\) be an admissible interval as in Definition 9. In this section we explain how to construct the BV operator on the filtered Floer cohomology: \[\label{eq:BV-filtered-appendix} \Delta:\mathit{HF}_{I}(H)\to \mathit{HF}_{I}(H).\tag{20}\] We remark that this “strict” filtration (i.e., that we can take the same interval \(I\) on both sides) uses the fact that \(H\) is autonomous.

As in Definition 8, let \(D_{t}\in \mathscr{O}(\epsilon)\) be an \(\epsilon\)-admissible perturbation. The BV operator is defined by counting the rigid solutions in the parametric moduli space of pairs \((\tau,u)\) such that: \[\label{eq:BV-equation} \left\{ \begin{align} &u:\mathbb{R}\times \mathbb{R}/\mathbb{Z}\to W,\quad \tau\in \mathbb{R}/\mathbb{Z},\\ &X_{\tau,s,t}=X_{H}+(1-\beta(s))X_{D_{t-\tau}}+\beta(s)X_{D_{t}}+\text{perturbation term},\\ &\partial_{s}u+J(u)(\partial_{t}u-X_{\tau,s,t}(u))=0,\\ &\lim_{s\to-\infty}u(s,t)=\gamma_{-}(t-\tau),\quad\lim_{s\to+\infty}u(s,t)=\gamma_{+}(t). \end{align} \right.\tag{21}\] The perturbation term should be very small (adding curvature at most \(\epsilon\)). Such a solution is interpreted as contributing to the coefficient \(\gamma_{+}\mapsto \gamma_{-}\). Counting the rigid elements defines a chain map: \[\Delta:\mathit{CF}(H+D_{t})\to \mathit{CF}(H+D_{t}).\]

The interpolation between \(X_{D_{t}}\) at the input and \(X_{D_{t-\tau}}\) at the output introduces some small amount of curvature, and so some care is needed to ensure the map on filtered homology is well-defined as in 20 .

This natural tranformation induces a well-defined map on the limits \(\mathit{HF}_{I}(H)\), following similar lines to Definition 12.

0.0.32 PSS elements↩︎

We discuss the construction of [51], [52] and its relation to the filtered Floer cohomology groups \(\mathit{HF}_{I}(H)\) for particular action windows \(I\).

Following [9], let us fix a cycle \(f:C\to W\) (a proper map) representing a cohomology class; e.g., the inclusion of a cotangent fiber into \(W=T^{*}M\) represents a degree \(n\) cohomology class. We also fix a Hamiltonian connection \(\mathfrak{H}\) (with connection potential \(\mathfrak{a}\)) as in the start of §[sec:sec:interf-with-tqft] on the cylinder with coordinates \(s+it\) satisfying:

• \(\mathfrak{a}=0\) for \(s\ge s_{0}\),

• \(\mathfrak{a}=H\mathit{d}t\) for \(s\le -s_{0}\).

From this data, the goal is to construct an element \(\mathit{PSS}_{\mathfrak{H}}(f)\in \mathit{HF}_{I}(H)\) for a suitable action window \(I\) depending on the connection \(\mathfrak{H}\), and show that the resulting element is independent of the choice of \(f\) up to proper cobordism.

We recall that a solution of the PSS equation for \((\mathfrak{H},J,f)\) is a pair \((x,u)\) such that \(u\) solves Floer’s equation for \((\mathfrak{H},J)\) and such that \(\lim_{s\to\infty}u(s,t)=f(x)\). Such solutions have a well-defined Fredholm theory when one compactifies the positive end as a removable singularity via the map \(\mathbb{R}\times \mathbb{R}/\mathbb{Z}\to \mathbb{C}\) sending \(s+it\) to \(e^{-2\pi(s+it)}\). We have the analogue of Definition 11.

In this section we discuss the compatibility of the structures from §[sec:sec:interf-with-tqft] with each other.

0.0.33 Rotationally symmetric continuations and the BV operator↩︎

Suppose that \(\mathfrak{H}\) is a Hamiltonian connection from \(H_{0}\) to \(H_{1}\) and the intervals \(I_{0},I_{1}\) are admissible so that \(\mathfrak{c}_{\mathfrak{H}}:\mathit{HF}_{I_{0}}(H_{0})\to \mathit{HF}_{I_{1}}(H_{1})\) is as in Lemma 31. Suppose additionally that \(\mathfrak{H}\) is rotationally symmetric in the sense that its connection potential \(\mathfrak{a}\) satisfies: \[\mathfrak{a}=K_{s}\mathit{d}s+H_{s}\mathit{d}t,\] i.e., there is no \(t\)-dependence. The standard example of such a connection considered in the body of the text is: \[K_{s}=0\quad\text{and}\quad H_{s}=\beta(s)H_{0}+(1-\beta(s))H_{1}.\]

0.0.34 Connections on the pair-of-pants via one-forms↩︎

In this section we explain a mechanism for constructing connections on the pair-of-pants using real-valued one-forms. The idea appears in the work of [3], [29]. The resulting product operations interact nicely with certain continuation maps.

A useful lemma about this is that the curvature term for such a connection can be computed explicitly:

Similarly we define continuation cylinders of type S:

It is more-or-less clear how continuation cylinders of type S can be “glued” to pairs-of-pants of type S, provided the function \(H\) is the same and the weights match at the interface. We have the following result ensuring that the resulting operations on \(\mathit{HF}_{I}\) compose in a similar manner. Due to the rather subtle nature of the product operation in filtered Floer cohomology §0.0.29, the statement is a bit technical. Nonetheless, the statement plays a crucial role in showing that the Viterbo restriction map of §[sec:sec:viterbo-restriction] respects the BV algebra structure.

0.0.35 Connections of type S, PSS elements, and the BV operator↩︎

In this section we continue with the special class of connections of type S introduced in §0.0.34.

Fix a connection \(\mathfrak{H}\) of type S on the cylinder for smooth function \(H\) with weights \((0,w')\) with \(w'>0\); this is a special case of Definition 16. Given \(f:C\to W\), we can apply the construction of §0.0.32 to produce an element: \[\mathit{PSS}(\mathfrak{H},f,J)\in \mathit{HF}_{w'I}(w'H)\] provided that \(I=[0,\tau]\) and \(\left\{0,\tau w'\right\}\not\in \mathit{Spec}(w'H)\). We have:

These type S PSS elements are compatible with type S continuation maps:

References↩︎

---

1. For odd-dimensional spheres, one can appeal to formulas in [12].↩︎

2. A manifold \(L\) is said to be aspherical if its universal cover is contractible.↩︎

3. Every connected manifold has a classifying map \(M\to BG\) where \(G=\pi_{1}(M,\mathit{pt})\); if the induced map in rational homology kills \([M]\) (we assume \(M\) is orientable so \([M]\) is well-defined), then we say \(M\) is rationally inessential; otherwise \(M\) is said to be rationally essential. Rationally inessential manifolds contain all simply connected manifolds, and form an ideal with respect to Cartesian product (in the space of orientable manifolds).↩︎

4. To check \(K_{\delta}\subset Q\times D(a)\), observe that: \[\max\left\{r_{1},r_{2}\right\}\le cf(c^{-1}r_{1})+r_{2}\text{ for all }c>0\] since \(f\) is convex and \(f(x)=x\) for \(x\ge 1\).↩︎

5. Standard continuation maps are defined by counting the solutions of infinitesimally small perturbations of the linear interpolation from \(H+D_{t}\) to \(H+D_{t}'\), as in [47].↩︎

6. There is an arbitrarily small additional error due to the fact we must add an infinitesimally small perturbation to the formula in 15 (the size can be taken to be as small as desired).↩︎