2.1 Length filtration

In this preparatory section we discuss the compatibility of our algebraic structures with suitable length filtrations. Let us fix a Riemannian metric on M and denote by $\Lambda^{<a}\subset\Lambda^{\leq a}\subset\Lambda$ the subspaces of loops of length $<a$ respectively $\leq a$ . For $a<b$ we set

\begin{equation*}H_*^{(a,b)}\Lambda := H_*(\Lambda^{<b},\Lambda^{\leq a}),\quad H^*_{(a,b)}\Lambda := H^*(\Lambda^{<b},\Lambda^{\leq a}).\end{equation*}

These groups form a double filtration in the sense that we have canonical maps $H_*^{(a,b)}\Lambda\to H_*^{(a',b')}\Lambda$ for $a\leq a'$ and $b\leq b'$ with the obvious properties, and similarly on cohomology.

For ${\varepsilon}\gt 0$ smaller than the length of the shortest closed geodesic on M the long exact sequence (1.2) becomes (by [Reference Cieliebak and OanceaCO18] and Poincaré duality on M) as

(2.1)

where the bottom row is the Gysin sequence of the sphere bundle $p:S^*M\to M$ with Euler class [e] and $\scriptstyle\mathcal{O}$ is the orientation local system on M. The products on $H_*^{(-{\varepsilon},{\varepsilon})}\Lambda$ and $\widehat{H}_*^{(-{\varepsilon},{\varepsilon})}\Lambda$ translate into the cup products on $H^{n-*}(M)$ and $H^{n-*}(S^*M)$ for which $p^*$ is a ring map. Poincaré duality becomes

where the bottom horizontal arrow is the classical Poincaré duality isomorphism. This supports our previous claim that the Poincaré duality, Theorem 1.1, should be seen as generalizing Poincaré duality for $S^*M$ (which is always an orientable manifold), and not as generalizing Poincaré duality on M itself.

All the results in this subsection have obvious counterparts for the based loop space ${\Omega}$ .

2.2 Critical levels

The length filtrations of the previous subsection yield an increasing filtration on $H_*\Lambda$ by

$$ H_*^{(-\infty,a)}\Lambda = H_*(\Lambda^{<a}),$$

and a decreasing filtration on $H^*\Lambda$ by

$$ H^*_{(a,\infty)}\Lambda = H^*(\Lambda,\Lambda^{\le a}).$$

We now consider an extension of the previous theorem to the setting of $\widehat{H}_*$ and $\widehat{H}^*$ . Recall from Theorem 2.1 that our choice of Riemannian metric determines an increasing filtration $\widehat{H}_*^{(-\infty.a)}\Lambda$ on $\widehat{H}_*\Lambda$ , and a decreasing filtration $\widehat{H}^*_{(a,\infty)}\Lambda$ on $\widehat{H}^*\Lambda$ . The following definition is analogous to Definition 2.3 above.

Recall that we denote the product on $\widehat{H}_*\Lambda$ by $\widehat{\bullet}$ and the product on $\widehat{H}^*\Lambda$ by $\widehat\circledast$ . The following extension of Theorem 2.4 is now an immediate consequence of the compatibility of these products with the filtrations in Theorem 2.1.

Note that each of the statements (1) and (2) is a consequence of the other one via the filtered Poincaré duality Theorem 2.1. Moreover, the Splitting Theorem 1.5 ensures that, except in the homological range $\{0,\dots,n\}$ , Theorem 2.4 is a consequence of either of the statements (1) or (2) in Theorem 2.6.

2.3 Hopf–Freudenthal–Gysin formulas

We refer the reader to the historical noteFootnote ⁷ for an explanation of our terminology.

Here, we discuss the relations between the product structures on the free loop space $\Lambda$ and on the based loop space ${\Omega}={\Omega}_{q}M$ . We denote the Pontrjagin product (of degree 0) on $H_*{\Omega}$ by $\bullet_{\Omega}$ , and the based cohomology product (of degree $n-1$ ) on $H^*({\Omega},q)$ (relative to the constant loop q at the basepoint) by $\circledast_{\Omega}$ . Recall that the inclusion $i:{\Omega}\hookrightarrow\Lambda$ induces pushforward/pullback maps $i_*:H_*{\Omega}\to H_*\Lambda$ and $i^*:H^*\Lambda\to H^*{\Omega}$ , as well as shriek maps (induced by intersection with the codimension n submanifold ${\Omega}\subset\Lambda$ ) $i_!:H_*\Lambda\to H_{*-n}{\Omega}$ and $i^!:H^*{\Omega}\to H^{*+n}\Lambda$ .

The above theorem extends to $\widehat{H}_*$ and $\widehat{H}^*$ as follows. We denote the products on $\widehat{H}_*\Lambda$ by $\widehat\bullet$ (of degree $-n$ ), on $\widehat{H}_*\Omega$ by $\widehat\bullet_{\Omega}$ (of degree 0), on $\widehat{H}^*\Lambda$ by $\widehat\circledast$ (of degree $n-1$ ), and on $\widehat{H}^*\Omega$ by $\widehat\circledast_\Omega$ (of degree $n-1$ ). The graded open–closed TQFT structures on $(\widehat{H}_*\Lambda,\widehat{H}_*{\Omega})$ and $(\widehat{H}^*\Lambda,\widehat{H}^*{\Omega})$ from Theorem 1.3 include in particular operations

\begin{gather*} {\boldsymbol{i}}_!:\widehat{H}_*\Lambda\to \widehat{H}_{*-n}\Omega,\quad {\boldsymbol{i}}^!:\widehat{H}^*\Omega\to \widehat{H}^{*+n}\Lambda,\cr {\boldsymbol{i}}_*:\widehat{H}_*\Omega\to \widehat{H}_*\Lambda,\quad {\boldsymbol{i}}^*:\widehat{H}^*\Lambda\to \widehat{H}^*\Omega.\end{gather*}

Here, ${\boldsymbol{i}}_!$ is the closed–open map defined by the zipper, ${\boldsymbol{i}}_*$ is the open–closed map defined by the cozipper, and ${\boldsymbol{i}}^!,{\boldsymbol{i}}^*$ are their algebraic duals. See

Figure 1.

Figure 1.

The zipper and the cozipper.

Figure 1 Long description

The image depicts two abstract shapes labeled as the zipper and the cozipper. The zipper is represented by a cylindrical shape with a split at the bottom, resembling a zipper opening. The cozipper is depicted as a bottle-like shape with two openings at the top, symbolizing a dual or complementary structure to the zipper. Both shapes have a plus sign at the top, indicating a positive or additive aspect.

Proof. Part (a) is an immediate consequence of the graded open–closed TQFT structure on $(\widehat{H}_*\Lambda,\widehat{H}_*{\Omega})$ from Theorem 1.4: the first relation says that the zipper is an algebra map, which is condition (3) in Definition 6.1, and the second relation says that the cozipper intertwines the canonical right module structures, which is derived from the axioms of a graded open–closed TQFT in [Reference Cieliebak and OanceaCO24c, Lemma 6.6]. See Figure 2 for picture proofs. Similarly, part (b) is a consequence of the graded open–closed TQFT structure on $(\widehat{H}^*\Lambda,\widehat{H}^*{\Omega})$ .

Figure 2.

TQFT proof of the Hopf–Freudenthal–Gysin formulas.

Figure 2 Long description

The image contains a series of diagrams that visually represent the mathematical relationships and operations involved in the proof of the Hopf-Freudenthal-Gysin formulas within the framework of topological quantum field theory. The diagrams depict various stages of the process, showing how different components interact and transform. Each diagram includes labeled elements such as plus and minus signs, indicating the orientation or charge of the components. The diagrams are interconnected, illustrating the flow and interaction of these components through different stages of the proof.

According to Theorem 1.4, statements (a) and (b) in Theorem 2.8 imply one another via Poincaré duality. Theorem 2.8 implies Theorem 2.7 via the inclusions in Theorem 1.4 and their based loop counterparts, using the fact that the maps ${\boldsymbol{i}}_!,{\boldsymbol{i}}^!,{\boldsymbol{i}}_*,{\boldsymbol{i}}^*$ agree on $H_*$ and $H^*$ with the topological shriek and pushforward/pullback maps, see § 6.3.

2.5 Manifolds all of whose geodesics are closed

Manifolds all of whose geodesics are simple and closed of the same primitive length have a long history of study, see [Reference BesseBes78]. In [Reference Goresky and HingstonGH09] it was observed that the loop and cohomology products have special properties on such manifolds.

A common generalization of this pair of results arises from the following special case of a theorem of P. Uebele on Rabinowitz Floer homology.

Then b is even and the following hold with coefficients in a field $\mathbf{k}$ .

1. (a) The class s is invertible and makes Rabinowitz Floer homology $SH_*({\partial} V)$ a free and finitely generated module over the ring of Laurent polynomials $\mathbf{k}[s,s^{-1}]$ .

2. (b) This module is (not necessarily freely) generated by classes that can be expressed as linear combinations of Reeb orbits of period at most T.

3. (c) $SH_*({\partial} V)$ and $SH_*(V)$ are finitely generated as $\mathbf{k}$ -algebras.

Proof of Corollary 2.14. We apply Theorem 2.12 to the unit disk cotangent bundle $V=D^*M$ . Note that $2c_1(D^*M)=0$ and the square of the canonical line bundle carries a canonical trivialization determined by its canonical structure of Lagrangian fibration. In this trivialization the Conley–Zehnder index equals the Morse index, see also Appendix A.1., and all closed geodesics have Conley–Zehnder index ( $=$ Morse index) $\gt 3-n$ by assumption (this is automatic if $n\gt 3$ because the Morse index is $\ge 0$ , and excludes the case $n=1$ , hence $S^*M$ is necessarily connected). Moreover, $b=n-1+\lambda\gt 0$ , where $\lambda\gt 3-n$ is the Morse index of a principal closed geodesic. So the hypotheses of Theorem 2.12 are satisfied, and parts (a) and (b) follow immediately.

Let $\scriptstyle\mathcal{O}$ be the orientation local system on M. For part (c), we refer to the Gysin sequence (2.1) and consider the induced splitting

(2.2) \begin{equation}H^{n-*}(S^*M) = \overline H^{n-*}(M)\oplus \overline H^{1-*}(M;\scriptstyle\mathcal{O}),\end{equation}

on the constant loops, where the first summand $\overline H^{n-*}(M)\simeq \overline H_*(M;\scriptstyle\mathcal{O})$ is contained in $\overline H_*\Lambda$ and the second one in $\overline H^{1-*}\Lambda$ . Denote by $H_*^{(0,\ell]}\Lambda\subset\widehat{H}_*\Lambda$ the subspace generated by the positively traversed Reeb orbits of period (or action) in $(0,\ell]$ . We claim that $\overline H_*\Lambda$ is the $\mathbf{k}[s]$ -submodule of $\widehat{H}_*\Lambda$ generated by the $\mathbf{k}$ -vector space

$$ V_* := \overline H^{n-*}(M)\oplus H_*^{(0,\ell]}\Lambda.$$

To see this we use that, by construction of the splitting, the summand $\overline H_*\Lambda$ is generated by the positively traversed Reeb orbits and the constant orbits generating $\overline H^{n-*}(M)$ , while $\overline H^*\Lambda$ is generated by the negatively traversed Reeb orbits and the constant orbits generating $\overline H^*(M;\scriptstyle\mathcal{O})$ . In particular, $s\in\overline H_*\Lambda$ . Since $\overline H_*\Lambda\subset\widehat{H}_*\Lambda$ is a subring, this proves the inclusion $\mathbf{k}[s]V_*\subset \overline H_*\Lambda$ . For the converse inclusion, we use an argument from [Reference UebeleUeb19]. By Theorem 2.12(b), the $\mathbf{k}$ -vector space $\widehat{H}_*\Lambda$ is generated by elements of the form $s^ka$ with $k\in{\mathbb{Z}}$ and $a\in H_*^{(0,\ell]}\Lambda$ . If $s^ka\in\overline H_*\Lambda$ , then by action reasons we must have $k\geq -1$ . If $k\geq 0$ , then $s^ka\in\mathbf{k}[s]H_*^{(0,\ell]}\Lambda\subset\mathbf{k}[s]V_*$ . If $k=-1$ , then $s^ka$ must belong to the constant part $\overline H^{n-*}(M)\subset V_*$ and the claim is proved.

By the claim, $\overline H_*\Lambda$ is finitely generated as a $\mathbf{k}[s]$ -submodule. It is torsion free because $\widehat{H}_*\Lambda$ is torsion free as a $\mathbf{k}[s,s^{-1}]$ -module. Since $\mathbf{k}[s]$ is a principal ideal domain, it follows that the $\mathbf{k}[s]$ -module $\overline H_*\Lambda$ is free. This proves the assertion on $\overline H_*\Lambda$ . An analogous argument gives the assertion on $\overline H^{1-*}\Lambda$ , which is the $\mathbf{k}[s^{-1}]$ -submodule of $\widehat{H}_*\Lambda$ generated by the $\mathbf{k}$ -vector space

$$V^* := H^{1-*}_{(0,\ell]}\Lambda\oplus \overline H^{1-*}(M;\scriptstyle\mathcal{O}).$$

Part (d) is an immediate consequence of part (c).

Corollary 2.14 requires neither that the geodesics have the same primitive length, nor that they are simple. In order to describe the algebra structure in examples, suppose now that all geodesics are closed with the same primitive length $\ell$ . Then the spaces $V_*$ and $V^*$ in the proof of Corollary 2.14 can be replaced by $\overline H_{n+*}(M;\scriptstyle\mathcal{O})\oplus s\,\overline H^{1-n-*}(M;\scriptstyle\mathcal{O})$ and $s^{-1}\overline H_{n+*}(M;\scriptstyle\mathcal{O})\oplus \overline H^{1-n-*}(M;\scriptstyle\mathcal{O})$ , respectively, and we extract from the proof the following statements.

• – The $\mathbf{k}[s,s^{-1}]$ -module $\widehat{H}_*\Lambda$ is (not necessarily freely) generated by

  $$ H^{-*}(S^*M) = \overline H_{n+*}(M;\scriptstyle\mathcal{O})\oplus \overline H^{1-n-*}(M;\scriptstyle\mathcal{O}).$$

• – The $\mathbf{k}[s]$ -submodule $\overline H_{n+*}\Lambda$ is (not necessarily freely) generated by

  $$ \widetilde V_* := \overline H_{n+*}(M;\scriptstyle\mathcal{O})\oplus s\,\overline H^{1-n-*}(M;\scriptstyle\mathcal{O}).$$

• – The $\mathbf{k}[s^{-1}]$ -submodule $\overline H^{1-n-*}\Lambda$ is (not necessarily freely) generated by

  $$ \widetilde V^* := s^{-1}\overline H_{n+*}(M;\scriptstyle\mathcal{O})\oplus \overline H^{1-n-*}(M;\scriptstyle\mathcal{O}).$$

Let us introduce the degree shifted algebra

$$ \widehat{{\mathbb{H}}}_*\Lambda := \widehat{H}_{*+n}\Lambda = \overline H_{*+n}\Lambda\oplus \overline H^{1-n-*}\Lambda,$$

graded by the shifted degree $|\gamma|=\mathrm{CZ}(\gamma)-n$ . Then the product has degree zero and is graded commutative, and the class s above has degree

$$ |s| = b = n-1+\lambda\gt 0$$

where $\lambda$ is the Morse index of a principal closed geodesic.

Example 2.17 (Spheres). For the loop space of $S^n$ the loop product and the cohomology product have been computed in [Reference Cohen, Jones and YanCJY04, Reference Goresky and HingstonGH09], respectively. Corollary 2.14 provides a simple way to derive one product from the other in the case $n\geq 3$ . For this, note first that in this case each closed geodesic has index at least $\lambda=n-1\gt 3-n$ , so Corollary 2.14 is applicable with the generator s of shifted degree

$$ |s|=n-1+\lambda=2n-2.$$

Now we distinguish two cases.

The case $n\geq 3$ odd. In this case the $\mathbf{k}[s,s^{-1}]$ -module $\widehat{{\mathbb{H}}}_*\Lambda S^n$ is generated by the graded vector space

$$ H^{-*}(S^*S^n) = {\rm span}_\mathbf{k}\{1,a,b,ab\}$$

in degrees

$$ |1|=0,\quad |a|=-n,\quad |b|=1-n,\quad |ab|=1-2n$$

where 1,a generate the first summand and b,ab the second one in the splitting (2.2). For degree reasons there can be no nontrivial relations involving different powers of s in ${\rm span}_\mathbf{k}\{1,a,b,ab\}\otimes _\mathbf{k}\mathbf{k}[s,s^{-1}]$ and we conclude that

$$ \widehat{{\mathbb{H}}}_*\Lambda S^n = {\rm span}_\mathbf{k}\{1,a,b,ab\}\otimes _\mathbf{k}\mathbf{k}[s,s^{-1}],$$

as a $\mathbf{k}[s,s^{-1}]$ -module. The preceding discussion then gives

$$ {\mathbb{H}}_{*}\Lambda S^n = {\rm span}_\mathbf{k}\{1,a,u,au\}\otimes _\mathbf{k}\mathbf{k}[s],\quad u:=sb,\quad |u|=n-1,$$

as a $\mathbf{k}[s]$ -module, and

$$ {\mathbb{H}}^{1-2n-*}\Lambda S^n = s^{-1}{\rm span}_\mathbf{k}\{1,a,u,au\}\otimes _\mathbf{k}\mathbf{k}[s^{-1}],$$

as a $\mathbf{k}[s^{-1}]$ -module. Here, the reduced (co)homologies are the same because $\chi(S^n)=0$ . To determine the ring structure we use an input from [Reference Cohen, Jones and YanCJY04] (see also Example 2.10 above), where it is shown that the ring structure on ${\mathbb{H}}_{*}\Lambda S^n$ has only one additional relation $u^2=s$ , hence

$$ {\mathbb{H}}_{*}\Lambda S^n = \mathbf{k}[a,u]/\langle a^2\rangle = \Lambda[a,u],$$

as a $\mathbf{k}$ -algebra. Since any relation in $\widehat{{\mathbb{H}}}_*\Lambda S^n$ gives rise under multiplication by a large negative power of s to a relation in ${\mathbb{H}}_{*}\Lambda S^n$ and vice versa, it follows that

$$ \widehat{{\mathbb{H}}}_*\Lambda S^n =\Lambda[a,u,u^{-1}] $$

as a $\mathbf{k}$ -algebra. This in turn implies that

$$ {\mathbb{H}}^{1-2n-*}\Lambda S^n = u^{-1}\Lambda[a,u^{-1}] $$

as a $\mathbf{k}$ -algebra. Since the classes $u^{-1},u^{-1}a$ correspond to the constant loops, the cohomology relative to the constant loops becomes

$$ {\mathbb{H}}^{1-2n-*}(\Lambda S^n,\Lambda_0S^n) = u^{-2}\Lambda[a,u^{-1}] $$

as a $\mathbf{k}$ -algebra, in accordance with [Reference Goresky and HingstonGH09].

The case $n\geq 3$ even. If $\mathbf{k}$ has characteristic 2 the (co)homology rings are exactly as in the case n odd. Suppose now that $\mathbf{k}$ has characteristic $\neq 2$ . Then the $\mathbf{k}[s,s^{-1}]$ -module $\widehat{{\mathbb{H}}}_*\Lambda S^n$ is generated by the graded vector space

$$ H^{-*}(S^*S^n) = {\rm span}_\mathbf{k}\{1,c\},$$

in degrees

$$ |1|=0,\quad |c|=1-2n$$

where 1 generates the first summand and c the second one in the splitting (2.2). Again there can be no nontrivial relations and we conclude that

$$ \widehat{{\mathbb{H}}}_*\Lambda S^n = {\rm span}_\mathbf{k}\{1,c\}\otimes _\mathbf{k}\mathbf{k}[s,s^{-1}],$$

as a $\mathbf{k}[s,s^{-1}]$ -module,

$$ \overline{\mathbb{H}}_{*}\Lambda S^n = {\rm span}_\mathbf{k}\{1,b\}\otimes _\mathbf{k}\mathbf{k}[s],\quad b:=sc,\quad |b|=-1,$$

as a $\mathbf{k}[s]$ -module, and

$$ \overline{\mathbb{H}}^{1-2n-*}\Lambda S^n = s^{-1}{\rm span}_\mathbf{k}\{1,b\}\otimes _\mathbf{k}\mathbf{k}[s^{-1}],$$

as a $\mathbf{k}[s^{-1}]$ -module. Note that the non-reduced loop space homology

$$ {\mathbb{H}}_{*}\Lambda S^n = ({\rm span}_\mathbf{k}\{1,b\}\otimes _\mathbf{k}\mathbf{k}[s])\oplus \mathbf{k} a,\quad |a|=-n,$$

is not free as a $\mathbf{k}[s]$ -module because $sa=0$ . The ring structure is again determined in [Reference Cohen, Jones and YanCJY04] (see Example 2.10 above), where it is shown that

$$ {\mathbb{H}}_{*}\Lambda S^n = \mathbf{k}[a,b,s]/\langle a^2,ab,b^2,2sa\rangle,$$

as a $\mathbf{k}$ -algebra. (Here, the factor 2 can be dropped because it is invertible in $\mathbf{k}$ , but the homology as written also gives the correct answer for $\mathbf{k}$ replaced by ${\mathbb{Z}}$ .) From this we again deduce the $\mathbf{k}$ -algebras

\begin{gather*} \widehat{{\mathbb{H}}}_*\Lambda S^n = \mathbf{k}[b,s,s^{-1}]/\langle b^2\rangle = \Lambda[b,s,s^{-1}], \cr \overline{\mathbb{H}}_{*}\Lambda S^n = \Lambda[b,s] , \quad \overline{\mathbb{H}}^{1-2n-*}\Lambda S^n = s^{-1} \Lambda[b,s^{-1}] , \cr {\mathbb{H}}^{1-2n-*}(\Lambda S^n,\Lambda_0S^n) = s^{-2}\Lambda[b,s^{-1}] ,\end{gather*}

the last one in accordance with [Reference Goresky and HingstonGH09]. Here, we use the notation $\Lambda[b,s,s^{-1}]$ to designate the free commutative algebra with generators $b,s,s^{-1}$ , with b of odd degree and s of even degree, and similarly for $\Lambda[b,s]$ and $\Lambda[b,s^{-1}]$ .

2.6 String point invertibility and resonances for CROSS

Example 2.17 can be generalized to all compact rank one symmetric spaces (CROSS), i.e. the projective spaces ${\mathbb{R}} P^n$ , ${\mathbb{C}} P^n$ , ${\mathbb{H}} P^n$ , and the Cayley plane $CaP^2$ . This is carried out in forthcoming joint work with E. Shelukhin where we compute for each CROSS the Rabinowitz loop homology ring together with its BV operator, and thus by restriction the BV algebra structures on its loop homology and loop cohomology. Moreover, we apply these computations to the following two questions.

2.6.1 String point invertibility

Consider a closed manifold M and denote by $\{\cdot,\cdot\}$ the Chas–Sullivan loop bracket on $H_*\Lambda$ .Footnote ⁸ For any given class $a\in H_*\Lambda$ , consider the operator $P_a:H_*(M)\to H_{*+\deg\, a -n+1}(M)$ defined by

$$ P_a=\mathrm{ev}_* \circ \{\cdot,a\}\circ i_*,$$

with $i:M\hookrightarrow \Lambda$ the inclusion of constant loops, and $\mathrm{ev}:\Lambda\to M$ the evaluation. We call M string point invertible if there exists a coefficient field $\mathbf{k}$ such that M is $\mathbf{k}$ -orientable, and a collection of classes $a_1,\dots,a_N\in H_*\Lambda$ such that

$$ [M]=P_{a_N}\circ\dots\circ P_{a_1}([pt]).$$

This property was introduced by Shelukhin [Reference ShelukhinShe22], who used it to prove the following conjecture of Viterbo in the case of string point invertible manifolds: The spectral norm of the pair consisting of the zero-section inside $T^*M$ and its image under a Hamiltonian diffeomorphism supported in the unit disc bundle is uniformly bounded. Shelukhin proved that spheres are string point invertible, and also that string point invertibility is preserved under taking products. Generalizing this, we will prove the following.

Theorem 2.18

1. (a) Let M be a CROSS modelled on ${\mathbb{C}} P^d$ , ${\mathbb{H}} P^d$ , or $CaP^2$ (set $d=2$ in this last case). Then M is string point invertible if and only if the Euler characteristic $\chi(M)=d+1$ is prime (with coefficient field $\mathbf{k}={\mathbb{Z}}/(d+1){\mathbb{Z}})$ .

2. (b) Let M be a CROSS modelled on ${\mathbb{R}} P^n$ , $n\ge 3$ . Then M is not string point invertible with ${\mathbb{Z}}/2$ -coefficients.

2.6.2 Resonances

Consider a closed Riemannian manifold M and fix a coefficient ring R. We associate with (co)homology classes $X\in H_k(\Lambda M)$ , $x\in H^k(\Lambda M)$ their degrees $\deg(X)=k$ , $\deg(x)=k$ and critical levels $\mathrm{ Cr}\, (X)$ , $\mathrm{ Cr}\, (x)$ as defined in § 2.2. We say that M is resonant with R-coefficients if there exists a constant $\overline\alpha\gt 0$ such that

$$ \deg(X) - \overline\alpha\,\mathrm{ Cr}\, (X)\quad \text{and}\quad \deg(x)-\overline\alpha\,\mathrm{ Cr}\, (x),$$

are uniformly bounded for all $X\in H_*(\Lambda M;R)$ and $x\in H^*(\Lambda M;R)$ . This property is introduced in [Reference Hingston and RademacherHR13] and its implications for indices and lengths of closed geodesics are discussed. Moreover, it is proved in [Reference Hingston and RademacherHR13] that spheres of dimension at least 3 are resonant with field coefficients. Generalizing this, we will prove the following.

Theorem 2.19. The string point invertible CROSS from the statement of Theorem 2.18(a) are resonant with coefficients in the field ${\mathbb{Z}}/(d+1){\mathbb{Z}}$ .

2.7 Index growth

Consider the following result on the index growth of an iterated closed geodesic.

Theorem 2.20 (Goresky and Hingston [Reference Goresky and HingstonGH09, Proposition 6.1]) Let $\gamma$ be a closed geodesic with index $\lambda$ and (transverse) nullity $\nu$ on a manifold of dimension n. Let $\lambda_m$ and $\nu_m$ denote the index and nullity of the m-fold iterate $\gamma^m$ . Then $\nu_m\le 2(n-1)$ and

(2.3) \begin{align} |\lambda_m-m\lambda| & \le (m-1)(n-1),\end{align}

(2.4) \begin{align}|\lambda_m+\nu_m-m(\lambda+\nu)| & \le (m-1)(n-1).\end{align}

These inequalities follow from standard properties of the Bott function $S^1\to {\mathbb{Z}}$ determined by the linearization of the geodesic flow along $\gamma$ , see [Reference BottBot56, Reference Goresky and HingstonGH09]. In the context of the present paper, we wish to explain that (2.3) and (2.4) are dual statements. We proceed as in the previous sections: first generalize each of these inequalities to a symplectic setting, then prove a duality theorem for the generalized statements.

The linearization of the geodesic flow along $\gamma$ determines a path in $\mathrm{Sp}(2(n-1))$ starting at Id, canonically defined up to conjugation. Based on Bott [Reference BottBot56], Long [Reference LongLon02] developed an index iteration theory for general paths of symplectic matrices, not necessarily obtained as linearizations of geodesic flows. To any path $P:[0,1]\to \mathrm{Sp}(2N)$ such that $P(0)=\mathrm{Id}$ is assigned its Bott–Long index

$$i(P)\in{\mathbb{Z}}.$$

See [Reference LongLon02, Definitions 5.2.7 and 5.4.1]. (In the notation of Long [Reference LongLon02] we have $i(P)=i_1(P)$ .) This is defined to be the Conley–Zehnder index of the concatenation $P\# P(1)\xi$ , where $\xi$ is a ‘minus curve’, i.e. a path of the form $\{t\mapsto \exp(tJS)\, : \, t\in[0,\varepsilon]\}$ with S a symmetric negative definite matrix and $\varepsilon\gt 0$ small.Footnote ⁹ The nullity of such a path is

$$\nu(P)=\nu(P(1))=\dim\ker (P(1)-\mathrm{Id}).$$

The key property that is of interest to us regarding the Bott–Long index is that, if $P=P_\gamma:[0,1]\to\mathrm{Sp}(2(n-1))$ is the linearized transverse geodesic flow along a given geodesic $\gamma$ , then $i(P_\gamma)$ equals the Morse index of $\gamma$ , cf. [Reference LongLon02, Theorem 7.3.4] and [Reference BottBot56, Theorem A]. Similarly, $\nu(P_\gamma)$ equals the nullity of $\gamma$ (in the transverse direction).

To formulate the iteration inequalities for the Bott–Long index, define $P^m:[0,m]\to \mathrm{Sp}(2N)$ by

$$P^m(t)=P(t-j)P(1)^j,\quad j\le t\le (j+1),\quad j=0,\dots,m-1.$$

If $P=P_\gamma:[0,1]\to\mathrm{Sp}(2(n-1))$ is the linearized transverse geodesic flow along a given geodesic $\gamma$ , then $P^m$ is the linearized transverse geodesic flow along the mth iterate $\gamma^m$ .

The following generalization of Theorem 2.20 is just a reformulation of a result by Liu and Long. It specializes to Theorem 2.20 if $P=P_\gamma$ is the linearized transverse geodesic flow along some geodesic $\gamma$ .

Theorem 2.21 (Liu and Long Long [Reference Liu and LongLL00]). Let $P:[0,1]\to \mathrm{Sp}(2N)$ be a continuous path with $P(0)=\mathrm{Id}$ . Then for all $m\ge 1$ we have

(2.5) \begin{align} |i(P^m)-m i(P)| & \le (m-1)N,\end{align}

(2.6) \begin{align} | (i(P^m)+\nu(P^m)) - m(i(P)+\nu(P)) | & \le (m-1)N.\end{align}

Proof. According to [Reference Liu and LongLL00, Theorem 1.2] (see also [Reference LongLon02, Theorem 10.1.3]), the following inequalities hold for all $m\ge 1$ :

\begin{align*}&m(i(P)+\nu(P)-N) +N -\nu(P)\\&\quad \le i(P^m) \\&\quad \le m(i(P)+N)-N-(\nu(P^m)-\nu(P)).\end{align*}

Theorem 2.21 follows directly from these using the additional obvious inequalities $2N\ge \nu(P^m)\ge \nu(P)\ge 0$ . For example, the second inequality implies

\begin{equation*} i(P^m)-m i(P) \le (m-1)N - (\nu(P^m)-\nu(P)) \le (m-1)N.\end{equation*}

The key definition for the duality statement is the following.

Definition 2.23. Given a path $P:[0,1]\to \mathrm{Sp}(2N)$ with $P(0)=\mathrm{Id}$ , the reverse path $\overline{P}:[0,1]\to \mathrm{Sp}(2N)$ is defined by

$$\overline{P}(t)=P(1-t) P(1)^{-1},\quad t\in[0,1].$$

Note that $\overline{P}(0)=\mathrm{Id}$ . The motivation for the definition is the following. Consider a 1-periodic compactly supported Hamiltonian $H:{\mathbb{R}}/{\mathbb{Z}} \times {\mathbb{R}}^{2N}\to {\mathbb{R}}$ and denote $\varphi_t$ , $t\in{\mathbb{R}}$ the flow of the Hamiltonian vector field $X_H^t$ , $t\in{\mathbb{R}}/{\mathbb{Z}}$ , which solves the equation $\frac{d}{dt}\varphi_t=X_H^t\circ \varphi_t$ with initial condition $\varphi_0=\mathrm{Id}$ . The 1-periodicity of the Hamiltonian implies $\varphi_t\circ\varphi_1=\varphi_1\circ\varphi_t=\varphi_{1+t}$ for all $t\in{\mathbb{R}}$ . The reverse flow $\overline{\varphi}_t=\varphi_{-t}$ satisfies the equation $\overline{\varphi}_t\circ\varphi_1=\varphi_{1-t}$ , and its linearization satisfies the equation $d\overline{\varphi}_t = d\varphi_{1-t}\circ d\varphi_1^{-1}$ . Thus, reversing the time direction for a Hamiltonian flow corresponds at the linearized level to reversal of the path as in Definition 2.23.

Proposition 2.24. Given a path $P:[0,1]\to \mathrm{Sp}(2N)$ with $P(0)=\mathrm{Id}$ , the index of the reverse path $\overline{P}$ is

(2.7) \begin{equation} i(\overline{P})=-i(P)-\nu(P).\end{equation}

This proposition is proved in Appendix B. Using it, we can state and prove our duality theorem for the index.

1. (i) The index inequality (2.5) for P is equivalent to the index+nullity inequality (2.6) for $\overline{P}$ .

2. (ii) The index+nullity inequality (2.6) for P is equivalent to the index inequality (2.5) for $\overline{P}$ .

Proof. Since taking the reverse of a path is an involutive operation, assertions (i) and (ii) are equivalent. To prove (ii) we use Proposition 2.24 and the equality $\overline{P}^m=\overline{P^m}$ to get

2.8 Level potency

Given a nonconstant closed geodesic c of length L, denote $Sc=S^{1}\cdot c \subset \Lambda$ its saturation with respect to the circle action and define its local level homology/cohomology

$$H_*(Sc) = \lim_{U\supset Sc} H_*(\Lambda \cap U,\Lambda ^{<L}\cap U),\quad H^*(Sc) = \lim_{U\supset Sc} H^{\ast }(\Lambda \cap U,\Lambda ^{<L}\cap U)$$

where U is an open set in $\Lambda$ containing Sc. Here, and in the rest of this subsection, we use rational coefficients.

For a closed orientable Riemannian n-manifold M, consider the following ensemble of dual results, essentially contained in [Reference HingstonHin93, Reference HingstonHin97, Reference Goresky and HingstonGH09].

We defer the proof of this theorem to Appendix C. Note that, by Theorem 2.20, part (a3) corresponds to the fastest possible growth of the index, while part (b3) corresponds to the slowest possible growth of the index + nullity under iteration.

Our goal is to generalize (a1–3) and (b1–3) to statements on Rabinowitz loop homology that are related by Poincaré duality. For this, consider $S^*M$ with its canonical contact form $\alpha$ . By a generalized closed Reeb orbit $\gamma$ we mean a closed Reeb orbit (i.e. the lift of a closed geodesic on M) or its backward parametrization. Recall that, together with the constants on $S^*M$ , after a Morse perturbation these are the generators of Rabinowitz Floer homology $SH_*(S^*M)=\widehat{H}_*\Lambda$ .

Let $\gamma$ be a generalized closed Reeb orbit of action $\int_\gamma\alpha = L\neq 0$ and $S\gamma=S^1\cdot\gamma$ its saturation. By local level Rabinowitz homology/cohomology, denoted $\widehat{H}_{\ast }(S\gamma)$ and $\widehat{H}^*(S\gamma)$ , we mean symplectic homology/cohomology of $S^*M$ localized near the isolated set $S\gamma$ in the following sense: we choose a Hamiltonian H as in the definition of $SH_*(S^*M)$ with negative slope $-\mu$ and positive slope $\tau$ such that $L\in (-\mu,\tau)$ , and consider local Floer (co)homology of the isolated fixed point set which corresponds to $S\gamma$ in the convex region of H. This is a mild variation on McLean’s definition of local symplectic homology in [Reference McLeanMcL12]. In case $\gamma$ is the lift of a nonconstant closed geodesic c, the Viterbo isomorphism (see [Reference ViterboVit96, Reference Abbondandolo and SchwarzAS10, Reference Cieliebak, Hingston and OanceaCHO23] and references therein) specializes to isomorphisms which intertwine the product structures

(2.8) \begin{equation}\widehat{H}_*(S\gamma)\simeq H_*(Sc), \quad \widehat{H}^*(S\gamma)\simeq H^*(Sc).\end{equation}

1. (A1) There exists $x\in \widehat{H}^*\Lambda$ such that $\mathrm{ Cr}\, (x^m)=m\mathrm{ Cr}\, (x)$ for all $m\ge 1$ .

2. (A2) There exists an isolated generalized closed Reeb orbit $\gamma$ and $x\in \widehat{H}^*(S\gamma)$ such that $x^m\in \widehat{H}^*(S\gamma^m)$ is nonzero for all $m\ge 1$ .

3. (A3) $S^*M$ carries an isolated generalized closed Reeb orbit $\gamma$ of index $\lambda$ with fastest possible index growth and $\widehat{H}^\lambda(S\gamma)\neq 0$ .

4. (B1) There exists $X\!\in\!\widehat{H}_*\Lambda$ such that $\mathrm{ Cr}\, \!(X^m)=m\mathrm{ Cr}\, \!(X)$ for all $m\ge 1$ .

5. (B2) There exists an isolated generalized closed Reeb orbit $\gamma$ and $X\in \widehat{H}_*(S\gamma)$ such that $X^m\in \widehat{H}_*(S\gamma^m)$ is nonzero for all $m\ge 1$ .

6. (B3) $S^*M$ carries an isolated generalized closed Reeb orbit $\gamma$ of index $\lambda$ and nullity $\nu$ with slowest possible index+nullity growth and $\widehat{H}_{\lambda+\nu+1}(S\gamma)\neq 0$ .

Let $\gamma$ be an isolated generalized closed Reeb orbit and $\overline\gamma$ its backward parametrization. Then (A2) for $\gamma$ is equivalent to (B2) for $\overline\gamma$ because Poincaré duality in Theorem 1.1 specializes to an isomorphism

\begin{equation*} \widehat{H}^{*}(S\gamma) \cong \widehat{H}_{1-*}(S\overline\gamma).\end{equation*}

Condition (A3) for $\gamma$ is equivalent to (B3) for $\overline\gamma$ : By Proposition 2.24, the indices and nullities are related by $i(\gamma)=-i(\overline\gamma)-\nu(\overline\gamma)$ , hence $\widehat{H}^{i(\gamma)}(S\gamma) \cong \widehat{H}_{1-i(\gamma)}(S\overline\gamma) = \widehat{H}_{i(\overline\gamma)+\nu(\overline\gamma)+1}(S\overline\gamma)$ . On the other hand, by the proof of Theorem 2.25, fastest possible index growth for $\gamma$ is equivalent to slowest possible index+nullity growth for $\overline\gamma$ .

To prove the equivalence with conditions (a1–3) and (b1–3), let $\gamma$ be the lift of an isolated closed geodesic c. Then by (2.8) and the above we have the following equivalences for ${i}=1,2,3$ :

\begin{gather*} \text{condition (a\textit{i}) for \textit{c}} \Longleftrightarrow \text{condition (A\textit{i}) for $\gamma$} \Longleftrightarrow \text{condition (B\textit{i}) for $\overline\gamma$},\cr \text{condition (b\textit{i}) for \textit{c}} \Longleftrightarrow \text{condition (B\textit{i}) for $\gamma$} \Longleftrightarrow \text{condition (A\textit{i}) for $\overline\gamma$}.\end{gather*}

Proposition 2.28 suggests a generalization of Theorem 2.27 in terms of local level Rabinowitz homology and closed Reeb orbits, which we will pursue in forthcoming work. It will rely on ideas from the proof of the Conley conjecture in [Reference HingstonHin09, Reference GinzburgGin10, Reference Ginzburg and GürelGG19] and the study of symplectically degenerate maxima in [Reference Ginzburg and GürelGG10, Reference Ginzburg, Hein, Hryniewicz and MacariniGHHM13, Reference Hein, Hryniewicz and MacariniHHM19].

3.1 Rabinowitz Floer homology

Recall from [Reference Cieliebak and FrauenfelderCF09] the definition of Rabinowitz Floer homology. Consider the completion $(\widehat V,\lambda)$ of a Liouville domain $(V,\lambda)$ with boundary $\Sigma={\partial} V$ . We abbreviate by $\mathcal{L}=C^\infty(S^1,\widehat V)$ the free loop space of $\widehat V$ , where $S^1={\mathbb{R}}/{\mathbb{Z}}$ . A defining Hamiltonian for $\Sigma$ is a smooth function $H:\widehat V\to{\mathbb{R}}$ with regular level set $\Sigma=H^{-1}(0)$ whose Hamiltonian vector field $X_H$ (defined by $i_{X_H}{\omega}=-dH$ ) has compact support and agrees with the Reeb vector field R along $\Sigma$ . Given such a Hamiltonian, the Rabinowitz action functional is defined by

$$ A_H: \mathcal{L} \times {\mathbb{R}} \to {\mathbb{R}}, \quad A_H(x,\eta):= \int_0^1 x^*\lambda - \eta \int_0^1 H(x(t))\, dt.$$

Critical points of $A_H$ are pairs $(x,\eta)$ such that $x:S^1\to \Sigma$ solves

\begin{equation*} \dot x = \eta X_H(x) = \eta R(x).\end{equation*}

There are three types of critical points: closed Reeb orbits on $\Sigma$ that are positively parametrized and correspond to $\eta\gt 0$ , closed Reeb orbits on $\Sigma$ that are negatively parametrized and correspond to $\eta<0$ , and constant loops on $\Sigma$ that correspond to $\eta=0$ . The action of a critical point $(x,\eta)$ is $A_H(x,\eta)=\eta$ . Because of general homotopy invariance properties of Floer homology, for the purpose of defining Rabinowitz Floer homology we can assume without loss of generality that the Reeb flow on $\Sigma$ is nondegenerate. In this case the critical points of $A_H$ form Morse-Bott manifolds: on the one hand $\Sigma$ , and on the other hand one circle for each closed Reeb orbit, parametrized positively or negatively.

Pick a smooth family $(J_t)_{t\in S^1}$ of compatible almost complex structures on $\widehat V$ that are cylindrical at infinity. It induces a metric $g=g_J$ on $\mathcal{L}\times \mathbb{R}$ which at a point $(x,\eta) \in \mathcal{L}\times \mathbb{R}$ and two tangent vectors $(\hat{x}_1,\hat{\eta}_1),(\hat{x}_2,\hat{\eta}_2)\in T_{(x,\eta)}(\mathcal{L}\times \mathbb{R})=\Gamma(S^1,x^*T\widehat V)\times \mathbb{R}$ is given by

$$ g_{(x,\eta)}((\hat{x}_1,\hat{\eta}_1),(\hat{x}_2,\hat{\eta}_2))=\int_0^1\omega(\hat{x}_1(t),J_t(x(t))\hat{x}_2(t))\, dt+\hat{\eta}_1\cdot\hat{\eta}_2.$$

Positive gradient flow lines of the Rabinowitz action functional $A_{H}$ with respect to this metric are solutions $(x,\eta)\in C^\infty(\mathbb{R}\times S^1,\widehat{V}) \times C^\infty(\mathbb{R},\mathbb{R})$ of the Rabinowitz Floer equation

(3.1) \begin{equation}\bigg\{\begin{array}{l}\partial_s x+J_t(x)(\partial_t x-\eta X_{H}(x))=0,\\\partial_s\eta + \int_0^1H(x(t))\, dt=0.\end{array}\end{equation}

We fix action values $-\infty<a<b<\infty$ outside the action spectrum of $A_H$ , and we pick an additional small Morse function f on the critical manifold ${\rm Crit}(A_H)$ . Thus f consists of a Morse function $f_\Sigma:\Sigma\to{\mathbb{R}}$ and Morse functions $f_{\gamma^k}:\mathrm{im}(\gamma)\to{\mathbb{R}}$ for each simple closed Reeb orbit $\gamma$ and $k\in{\mathbb{Z}}\setminus\{0\}$ , where we assume the latter to have unique minima $m_{\gamma^k}$ and maxima $M_{\gamma^k}$ .

We fix a commutative unital coefficient ring R. Then $RFC_*^{(a,b)}(H,f;J)$ is the free R-module generated by the critical points of f with action in (a,b), and the boundary operator

$$ {\partial}: RFC_*^{(a,b)}(H,f;J)\to RFC_{*-1}^{(a,b)}(H,f;J)$$

counts cascades as in [Reference Cieliebak and FrauenfelderCF09]. They combine the negative gradient flow of $A_H$ with respect to the metric g and the negative gradient flow of f with respect to some metric on ${\rm Crit}(A_H)$ . As grading we use the integer grading obtained by shifting up by $\frac 12$ the half-integer grading defined in [Reference Cieliebak and FrauenfelderCF09]. The resulting filtered Rabinowitz Floer homology groups

$$ RFH_*^{(a,b)}(\Sigma,\lambda):=RFH_*^{(a,b)}(H,f;J),$$

are well defined and do not depend on the choice of J, H, and f, although they do depend on the contact form $\lambda|_\Sigma$ . The Rabinowitz Floer homology of $\Sigma$ is defined as the limit

$$ RFH_*(\Sigma) := \lim_{\stackrel \longrightarrow b} \lim_{\stackrel \longleftarrow a} RFH_*^{(a,b)}(\Sigma,\lambda), \quad a\to-\infty,\ b\to\infty.$$

By [Reference Cieliebak and FrauenfelderCF11, Theorem A], this definition is equivalent to the original one in [Reference Cieliebak and FrauenfelderCF09]. By similar direct-inverse limits one defines as in [Reference Cieliebak and OanceaCO18] the groups

$$ RFH_*^\heartsuit(\Sigma),\quad \heartsuit\in\{\varnothing, \gt 0,\ge 0, =0, \le 0, <0\},$$

with the meaning that $RFH_*^\varnothing=RFH_*$ .

We define the Rabinowitz Floer cohomology groups by a similar procedure using the dual complex $RFC^*_{(a,b)}(H,f;J)=RFC_*^{(a,b)}(H,f;J)^\vee$ . The filtered Rabinowitz Floer cohomology groups are defined as

$$RFH^*_{(a,b)}(\Sigma,\lambda):= RFH^*_{(a,b)}(H,f;J),$$

and the Rabinowitz Floer cohomology of $\Sigma$ is the limit

$$RFH^*(\Sigma):= \lim_{\stackrel \longrightarrow a} \lim_{\stackrel \longleftarrow b} RFH^*_{(a,b)}(\Sigma,\lambda),$$

with variants $RFH^*_\heartsuit(\Sigma)$ for $\heartsuit\in\{\varnothing, \gt 0,\ge 0, =0, \le 0, <0\}$ . The grading is the same as for homology. This definition is analogous to [Reference Cieliebak and OanceaCO18, Definition 3.1], given in a symplectic homology framework.

These Rabinowitz Floer co/homology groups depend on the contact structure on $\Sigma$ (though not on the contact form) as well as the Liouville filling V (up to Liouville homotopy). As in [Reference Cieliebak and OanceaCO18] we do not indicate the filling in the notation since it will always be clear from the context.

3.2 Poincaré duality

Poincaré duality results from the following observation: under the canonical involution

$$ {\mathcal L}\times {\mathbb{R}} \to {\mathcal L}\times {\mathbb{R}}, \quad (x,\eta)\mapsto(\bar x,\bar\eta),\quad \bar x(t)=x(-t),\quad \bar\eta=-\eta,$$

the Rabinowitz action functional changes sign

$$ A_H(\bar x,\bar\eta) = -A_H(x,\eta).$$

It follows that the involution maps positive gradient lines of $A_H$ to negative gradient lines of $A_H$ , provided that we also replace the family $J_t$ by $\bar J_t:=J_{-t}$ (and the resulting metric accordingly). In other words, if $(x,\eta)\in C^\infty(\mathbb{R}\times S^1,\widehat{V}) \times C^\infty(\mathbb{R},\mathbb{R})$ solves the Rabinowitz Floer equation (3.1) with positive asymptotic $(x_+,\eta_+)$ and negative asymptotic $(x_-,\eta_-)$ , then $(\bar x,\bar\eta)$ defined by

$$ \bar x(s,t):=x(-s,-t),\quad \bar\eta(s):=-\eta(-s)$$

solves equations (3.1) with positive asymptote $(\bar x_-,\bar\eta_-)$ and negative asymptote $(\bar x_+,\bar\eta_+)$ .

When applying the involution, we also replace the Morse functions $f_\Sigma:\Sigma\to{\mathbb{R}}$ and $f_{\gamma^k}:\mathrm{im}(\gamma)\to{\mathbb{R}}$ by

$$ \bar f_\Sigma:=-f_\Sigma,\quad \bar f_{\gamma^k}:=-f_{\gamma^k}.$$

Then the preceding discussion shows that the involution $(x,\eta)\mapsto(\bar x,\bar\eta)$ defines a chain isomorphism between the Rabinowitz Floer chain and cochain groups

$$ RFC_*^{(a,b)}(H,f;J)\stackrel{\cong}\longrightarrow RFC^{1-*}_{(-b,-a)}(H,\bar f;\bar J).$$

Therefore, we have shown the following.

Theorem 3.1 (Poincarà duality in Rabinowitz Floer homology). The involution $(x,\eta)\mapsto(\bar x,\bar\eta)$ induces isomorphisms between filtered Rabinowitz Floer homology and cohomology groups

$$ PD: RFH_*^{(a,b)}(\Sigma,\lambda)\stackrel{\cong}\longrightarrow RFH^{1-*}_{(-b,-a)}(\Sigma,\lambda),$$

and between the Rabinowitz Floer homology and cohomology groups

$$ PD: RFH_*^\heartsuit(\Sigma)\stackrel{\cong}\longrightarrow RFH^{1-*}_{-\heartsuit}(\Sigma)$$

for $\heartsuit\in\{\varnothing, \gt 0,\ge 0, =0, \le 0, <0\}$ .

Given $\heartsuit\in\{\varnothing, \gt 0,\ge 0, =0, \le 0, <0\}$ , the meaning of $-\heartsuit$ is that equalities are preserved and inequalities are reversed, e.g., if $\heartsuit=\varnothing$ then $-\heartsuit=\varnothing$ , and if $\heartsuit=`\ge 0'$ then $-\heartsuit=`\le 0'$ .

4.1 Recollections on Poincaré duality and exact sequences

The main result in [Reference Cieliebak, Frauenfelder and OanceaCFO10] states that

(4.1) \begin{equation} RFH_*^\heartsuit({\partial} V) \cong SH_*^\heartsuit({\partial} V)\end{equation}

where V is a Liouville domain of dimension 2n and $SH_*^\heartsuit({\partial} V)$ denotes the symplectic homology of the trivial cobordism $[\frac 1 2,1]\times {\partial} V$ in the sense of [Reference Cieliebak and OanceaCO18]. We assume that $2c_1(V)=0$ and that the square of the canonical bundle of TV is trivialized, so that Floer chain complexes are canonically ${\mathbb{Z}}$ -graded by the Conley–Zehnder indices of periodic orbits, see Appendix A. This assumption is used only for grading purposes and has no essential bearing on the sequel arguments. Then Theorem 3.1 has the following formulation in symplectic homology.

Theorem 4.1 (Poincarà duality in symplectic homology of a trivial cobordism [Reference Cieliebak and OanceaCO18, Theorem 9.4]). There exist canonical isomorphisms between the symplectic homology and cohomology groups of a trivial cobordism

$$ PD: SH_*^\heartsuit({\partial} V)\stackrel{\cong}\longrightarrow SH^{1-*}_{-\heartsuit}({\partial} V)$$

for $\heartsuit\in\{\varnothing, \gt 0,\ge 0, =0, \le 0, <0\}$ .

This isomorphism is constructed in [Reference Cieliebak and OanceaCO18] without relying on the connection between symplectic homology and Rabinowitz Floer homology. We expect the isomorphisms of Theorems 3.1 and 4.1 to be compatible with (4.1). In the rest of the paper we will refer to Poincaré duality for a trivial cobordism as being the isomorphism in Theorem 4.1.

The Poincaré duality isomorphism has the following properties, see [Reference Cieliebak and OanceaCO18].

(A) Compatibility with the exact sequence of the pair $(V,{\partial} V)$ .

Theorem 4.2 [Theorem 9.5]. For every Liouville domain V and $\heartsuit\in\{\varnothing, \gt 0,\ge0, =0, \le 0, <0\}$ there exists a commuting diagram

(4.2)

where the rows are the long exact sequences of the pair $(V,{\partial} V)$ from [Reference Cieliebak and OanceaCO18] and the vertical arrows are the Poincaré duality isomorphisms from Theorem 4.1 (the third one) and for pairs as defined in [Reference Cieliebak and OanceaCO18] (the other ones). Moreover, the Poincaré duality isomorphisms are compatible with filtration exact sequences.

(B) Relation to singular cohomology. Recall from [Reference Cieliebak and OanceaCO18] that at action zero symplectic homology specializes to singular cohomology,

$$ SH_*^{=0}(V) \cong H^{n-*}(V),$$

and similarly for the other versions. Therefore, we obtain the following.

Corollary 4.3 [Reference Cieliebak and OanceaCO18, Corollary 9.7]. The commuting diagram in Theorem 4.2 specializes at action zero to

(4.3)

where the rows are the long exact sequences of the pair $(V,{\partial} V)$ and the vertical arrows are the Poincaré duality isomorphisms for the closed manifold ${\partial} V$ (the third one) and the manifold-with-boundary V (the other ones).

(C) Description of the first map in (4.2). The following result is simply a restatement of [Reference Cieliebak, Frauenfelder and OanceaCFO10, Proposition 1.3].

Given an exact n-dimensional Lagrangian L which is conical near its boundary ${\partial} L$ , the Poincaré duality isomorphisms from [Reference Cieliebak and OanceaCO18] read

$$SH_*(L,{\partial} L)\simeq SH^{n-*}(L),$$

and

(4.6) \begin{equation} SH_*({\partial} L)\simeq SH^{n-*+1}({\partial} L).\end{equation}

This fixes the grading for the Lagrangian counterpart of (4.2).

The action zero part of Lagrangian symplectic homology and cohomology is expressed in topological terms as

$$SH_*^{=0}(\mathcal{L})\simeq H^{n-*}(\mathcal{L}), \quad SH^*_{=0}(\mathcal{L})\simeq H_{n-*}(\mathcal{L})$$

where $\mathcal{L}$ stands for any of the symbols L, $(L,{\partial} L)$ , or ${\partial} L$ . This fixes the grading for the Lagrangian counterpart of (4.3).

The Lagrangian analogue of (4.4) is as follows.

In the particular case where L is a disc the bottom map vanishes for degree reasons, hence the map $SH_*(L,{\partial} L)\to SH_*(L)$ vanishes as well.

4.2 TQFT operations on Floer homology

In this subsection we recall the definition of TQFT operations on Hamiltonian Floer homology from [Reference SeidelSei08a, Reference RitterRit13], see also [Reference Cieliebak and OanceaCO18, Reference Ekholm and OanceaEO17]. Consider a punctured Riemann surface S with p negative and q positive punctures and chosen cylindrical ends $Z_i^-=(-\infty,0]\times S^1$ near the negative punctures $z_i^-$ , respectively $Z_j^+=[0,\infty)\times S^1$ near the positive punctures $z_j^+$ . Let $H:S\times \widehat V\to{\mathbb{R}}$ be an S-dependent Hamiltonian on a completed Liouville domain $\widehat V$ which is linear outside a compact subset of $\widehat V$ , and S-independent equal to $H_\ell^\pm$ near each puncture $z_\ell^\pm$ of S. Pick nonzero weights $A_\ell^\pm\in{\mathbb{R}}^*$ (one usually considers positive weights, but we will also need to allow negative ones) and a 1-form $\beta$ on S with the following properties:

1. (i) $d_S(H\beta)\leq 0$ ;

2. (ii) $\beta=A_\ell^\pm dt$ in cylindrical coordinates $(s,t)\in Z_\ell^\pm$ near the puncture $z_\ell^\pm$ .

We consider maps $u:S\to\widehat V$ that are perturbed holomorphic in the sense that $(du-X_H\otimes \beta)^{0,1}=0$ and have finite energy $E(u)=\frac{1}{2}\int_S|du-X_H\otimes \beta|^2{\rm vol}_S$ . They converge at the punctures to 1-periodic orbits $x_\ell^\pm$ of $A_\ell^\pm H_\ell^\pm$ and satisfy the energy estimate

(4.7) \begin{equation} 0 \leq E(u) \leq \sum_{j=1}^qA_{A_j^+H_j^+}(x_j^+) - \sum_{i=1}^pA_{A_i^-H_i^-}(x_i^-).\end{equation}

The signed count of such maps yields an operation

$$ \psi_S:\bigotimes _{j=1}^qFH_*(A_j^+H_j^+) \to \bigotimes _{i=1}^pFH_*(A_i^-H_i^-)$$

of degree $n(2-2g-p-q)$ which does not increase the action. These operations are graded commutative if the degrees are shifted by $-n$ and satisfy the usual TQFT composition rules. As they respect the action, the operations descend to operations between suitable filtered Floer homology groups and thus to symplectic homology.

Suppose now that $H:\widehat V\to{\mathbb{R}}$ is S-independent. Then $\beta$ and the weights are related by Stokes’ theorem

$$ \sum_{j=1}^qA_j^+ - \sum_{i=1}^pA_i^- = \int_Sd\beta.$$

Conversely, if the quantity on the left-hand side is non-negative (respectively zero, respectively non-positive), then we find a 1-form $\beta$ with properties (i) and (ii) such that $d\beta\geq 0$ (respectively $=0$ , respectively $\leq0$ ). Thus for S-independent H we can arrange conditions (i)–(ii) in the following situations:

1. (a) H arbitrary, $d\beta\equiv 0$ , $p,q\geq 1$ ;

2. (b) $H\geq 0$ , $d\beta\leq 0$ , $p\geq 1$ ;

3. (c) $H\leq 0$ , $d\beta\geq 0$ , $q\geq 1$ .

Let us now specialize to the case that S is a pair-of-pants with two positive punctures and one negative puncture. Then the operation $\psi_S$ induces a pair-of-pants product on filtered Floer homology

\begin{equation*}\psi_S:FH_i^{(a_1,b_1)}(A_1^+H)\otimes FH_j^{(a_2,b_2)} (A_2^+H) \longrightarrow FH_{i+j-n}^{(\max\{a_1+b_2,a_2+b_1\},b_1+b_2)}(A_1^-H).\end{equation*}

By taking suitable inverse and direct limits (see [Reference Cieliebak and OanceaCO18]) this leads to degree $-n$ pair-of-pants products on $SH_*(V)$ , $SH_*(V,{\partial} V)\cong SH^{-*}(V)$ , $SH_*({\partial} V)$ , and $SH_*(W,{\partial} W)\cong SH^{-*}({\partial} V)$ , where $W=[\frac12,1]\times {\partial} V$ .

More generally, one can consider open–closed TQFT operations in Floer homology, which mix together Hamiltonian Floer homology inputs and outputs and Lagrangian Floer homology inputs and outputs. The discussion about weights carries over without modification, and we refer to Appendix A for a discussion of gradings in this setting.

4.3 Products and the mapping cone

We consider in this section an algebraic set-up that will help us organize the subsequent geometric arguments regarding the compatibility of the Poincaré duality isomorphism with product structures.

Consider a chain complex $(C,{\partial})$ of the form

$$ C=C^-\oplus C^+,\quad {\partial} = \bigg(\begin{array}{c@{\quad }c}{\partial}^- & f \\0 & {\partial}^+ \end{array}\bigg).$$

Thus $(C^\pm,{\partial}^\pm)$ are chain complexes, $C^-$ is a subcomplex of C, and C is the cone of the chain map $f:C^+[1]\to C^-$ . We use the conventions of [Reference Cieliebak and OanceaCO18] for cones and degree shifts: $C^+[1]_i=C_{i+1}$ and ${\partial}_{C^+[1]}=-{\partial}^+$ .

Assume now that C is acyclic as a consequence of $\mathrm{Id}_C$ being homotopic to zero, i.e. there exists $K:C\to C[1]$ such that

$$ \mathrm{Id}_C={\partial} K + K{\partial}.$$

Writing K with respect to the decomposition $C=C^-\oplus C^+$ as

$$ K=\left(\begin{array}{c@{\quad }c}k^- & h \\g & k^+ \end{array}\right),$$

this is equivalent to the system of four equations

$$\left\{\begin{array}{l}{\partial}^+g+g{\partial}^- = 0, \\[2pt]{\partial}^-k^- + fg + k^-{\partial}^- = \mathrm{Id}_{C^-}, \\[2pt]{\partial}^+k^+ + gf + k^+{\partial}^+ = \mathrm{Id}_{C^+}, \\[2pt]{\partial}^-h + fk^+ + k^-f + h{\partial}^+ = 0.\end{array}\right.$$

The first three equations amount to the fact that

$$ g:C^-\to C^+[1]$$

is a chain map which is a homotopy inverse for f, the homotopies between fg and gf and the corresponding identity maps being given by

$$ k^-:C^-\to C^-[1],\quad k^+:C^+\to C^+[1].$$

The fourth equation gives extra information which we will not use.

Let now $(\overline C,\overline{\partial})$ be another chain complex of the form

$$ \overline C=\overline C^-\oplus \overline C^+,\quad \overline{\partial} = \bigg(\begin{array}{c@{\quad }c}\overline{\partial}^- & \overline f \\[2pt] 0 & \overline{\partial}^+ \end{array}\bigg).$$

The complex $\overline C$ need not be acyclic, although an important special case is $\overline C=C$ . In the following we will adopt the following conventions for a linear map $\phi:C\otimes C\to\overline C$ :

• – $\phi^{+-}_+$ is the part of $\phi$ mapping $C^+\otimes C^-\to \overline C^+$ , etc.;

• – we abbreviate $\phi^+:=\phi^{++}_+$ and $\phi^-:=\phi^{-}_-$ ;

• – $[{\partial},\phi]:=\overline{\partial}\circ\phi - (-1)^{|\phi|}\phi\circ({\partial}\otimes 1+1\otimes {\partial})$ .

Suppose now that we are given a ‘product’ $\mu:C\otimes C\to\overline C$ of degree zero satisfying

$$ [{\partial},\mu]=0 \quad \text{and}\quad \mu^{--}_+ = \mu^{-+}_+ = \mu^{+-}_+ = 0.$$

Thus $\mu$ descends to homology and the pair of subcomplexes $(C^-,\overline C^-)$ is a ‘two-sided ideal pair’ with respect to $\mu$ .Footnote ¹⁰ It follows that

$$ 0=[{\partial},\mu]^{-}_-=[{\partial}^-,\mu^{-}_-] \quad \text{and}\quad 0=[{\partial},\mu]^{++}_+=[{\partial}^+,\mu^{++}_+],$$

so $\mu^-=\mu^{-}_-:C^-\otimes C^-\to\overline C^-$ and $\mu^+=\mu^{++}_+:C^+\otimes C^+\to\overline C^+$ descend to products on homology.

The following lemma is a key result for this section.

1. (a) The negative parts $\nu^-,\widetilde\nu^-:C^-\otimes C^-\to\overline C^-[1]$ of the maps $\nu:=\mu(K\otimes 1):C\otimes C\to\overline C[1]$ and $\widetilde\nu:=\mu(1\otimes K)$ satisfy

   $$ \mu^- = [{\partial}^-,\nu^-] = [{\partial}^-,\widetilde\nu^-].$$
   In particular, the primary product $\mu^-$ vanishes on the homology of $C^-$ .

2. (b) The negative part $\sigma^-:C^-[-1]\otimes C^-[-1]\to\overline C^-[-1]$ of the difference $\sigma:=\widetilde\nu-\nu$ satisfies $[{\partial}^-,\sigma^-]=0$ and thus descends to a secondary product on the homology of $C^-[-1]$ .

3. (c) The negative part $\eta^-=\eta^{-}_-:C^-[-1]\otimes C^-[-1]\to\overline C^-$ of $\eta:=\mu(K\otimes K)$ satisfies

   $$ [{\partial}^-,\eta^-] = \sigma^-\overline f\mu^+(g\otimes g).$$
   Thus the secondary product $\sigma^-$ agrees on homology with the product $\mu^+$ transferred from $C^+$ to $C^-[-1]$ via the chain map $\overline f$ and the homotopy inverse g of f.

Proof. For part (a) we compute, using $[{\partial},\mu]=0$ and $[{\partial},K]=1$ ,

\begin{align*} [{\partial},\nu] &= \overline{\partial}\mu(K\otimes 1) + \mu(K\otimes 1)({\partial}\otimes 1+1\otimes {\partial}) \cr &= \mu([{\partial}\otimes 1+1\otimes {\partial},K\otimes 1]) \cr &= \mu([{\partial},K]\otimes 1) = \mu.\end{align*}

We now use $\nu^{-}_+=0$ and then take the $C^-$ part to get

$$ \mu^- = [{\partial},\nu]^{-}_- = [{\partial}^-,\nu^-].$$

Part (b) follows directly from part (a). For part (c) we compute, using $[{\partial},\mu]=0$ and $[{\partial},K]=1$ ,

\begin{align*} [{\partial},\eta] &= \overline{\partial}\mu(K\otimes K) + \mu(K\otimes K)({\partial}\otimes 1+1\otimes {\partial}) \cr &= \mu([{\partial}\otimes 1+1\otimes {\partial},K\otimes K]) \cr &= \mu([{\partial},K]\otimes K-K\otimes [{\partial},K]) \cr &= \mu(1\otimes K-K\otimes 1).\end{align*}

Taking the $C^-$ part the right-hand side becomes $\widetilde\nu^-\nu^-=\sigma^-$ , while the left-hand side becomes

$$ [{\partial},\eta]^{-}_- = [{\partial}^-,\eta^-] + \overline f\eta^{-}_+ = [{\partial}^-,\eta^-] + \overline f\mu^+(g\otimes g).$$

4.4 Poincaré duality with products

As in the previous sections V is a Liouville domain of dimension 2n, and $W=[\frac 1 2, 1]\times {\partial} V\subset V$ is the trivial cobordism realized by a collar neighbourhood of ${\partial} V$ in V. Recall from § 4.2 that $SH_*(W,{\partial} W)$ carries a product of degree $-n$ determined by counts of rigid pairs-of-pants in combination with suitable action truncations. We refer to this product as the primary product on $SH_*(W,{\partial} W)$ . By Poincaré duality we have $SH_*(W,{\partial} W)\simeq SH^{-*}(W)$ , and we refer to the corresponding degree n pair-of-pants product on $SH^*(W)$ as the primary product on $SH^*(W)=SH^*({\partial} V)$ .

The following result corresponds to Theorem 1.1 from the Introduction. Recall that $SH_*({\partial} V)$ carries a unital product ${\boldsymbol{\mu}}$ of degree $-n$ .

Part (b) shows that, in contrast to the secondary product on $SH^*_{\gt 0}(V)$ defined in [Reference Cieliebak and OanceaCO18], the secondary product on $SH^*({\partial} V)$ has a unit.

It is interesting to note that our proof of (a) is inseparable from the proof of (b). In particular, unitality of ${\boldsymbol{\lambda}}^\vee \tau$ is only implicitly inferred from unitality of ${\boldsymbol{\mu}}$ via the isomorphism in (b). We refer to [Reference Cieliebak and OanceaCO24a] for an alternative, more direct description of the unit for ${\boldsymbol{\lambda}}^\vee \tau$ within the framework of multiplicative structures on cones.

Before giving the proof of Theorem 4.8, we state its Lagrangian counterpart. We consider Maslov 0 exact Lagrangians $L\subset V$ with boundary such that ${\partial} L=L\cap {\partial} V$ and L is conical near ${\partial} L$ .

1. (a) The primary product on $SH^*({\partial} L)$ vanishes. As a consequence, $SH^*({\partial} L)$ carries a secondary product $\boldsymbol{\lambda}^\vee \tau$ of degree $-1$ , which has a unit in degree 1.

2. (b) The Poincaré duality isomorphism

   $$ PD:SH_*({\partial} L)\stackrel\simeq \longrightarrow SH^{n-*+1}({\partial} L)$$
   is a ring homomorphism, where $SH_*({\partial} L)$ is endowed with its degree $-n$ pair-of-pants product $\boldsymbol{\mu}$ and $SH^*({\partial} L)$ is endowed with its degree $-1$ secondary product $\boldsymbol{\lambda}^\vee \tau$ .

Before giving with the proof of Theorem 4.8, it is useful to recall the definition of $SH_*({\partial} V)$ . Denote the radial coordinate in the conical part of the symplectic completion $\widehat V=V\cup [1,\infty)\times {\partial} V$ by $r\in[1,\infty)$ . Rabinowitz Floer homology $SH_*({\partial} V)$ is defined using the family

$$\mathcal{H}=\{H_{\lambda,\mu}\, : \, \lambda,\mu\in{\mathbb{R}}\}$$

of Hamiltonians on $\widehat V$ , equal to 0 on ${\partial} V$ , linear in r of slope $\mu$ on $[1,\infty)\times {\partial} V$ , linear in r of slope $\lambda$ on $[1/2,1]\times {\partial} V$ , and constant equal to $-\lambda/2$ on $\{r\le 1/2\}$ . See Figure 3.

Figure 3.

Hamiltonian profile for $SH_*({\partial} V)$ .

Given $-\infty<a<b<\infty$ we define

$$SH_*^{(a,b)}({\partial} V)=\lim^{\longrightarrow}_{\mu\to\infty,\, \lambda\to-\infty}FH_*^{(a,b)}(H_{\lambda,\mu}),$$

and further

$$SH_*({\partial} V)=\lim^{\longrightarrow}_{b\to\infty}\lim^{\longleftarrow}_{a\to-\infty}SH_*^{(a,b)}({\partial} V).$$

Proof of Theorem 4.8. Let $\widehat V$ be the completion of V, with a canonical embedding of the symplectization $(0,\infty)\times {\partial} V\subset \widehat V$ . Under this embedding the level $\{1\}\times {\partial} V$ is canonically identified with ${\partial} V$ , and the restriction of this embedding to $(0,1]\times {\partial} V$ takes values in V. Denote $V_\delta= \widehat V\setminus (\delta,\infty)\times {\partial} V$ for $\delta\gt 0$ .

Key to the proof is the following construction. Given $\tau\gt 0$ define the function $\ell_{-\tau}:(0,\infty)\to{\mathbb{R}}$ constant equal to $3\tau/4$ on the interval $(0,\frac{1}{4}]$ and linear of slope $-\tau$ on $[\frac{1}{4},\infty)$ , so that $\ell_{-\tau}(1)=0$ . Fix now parameters $\tau'\gt \tau\gt 0$ and $0<\epsilon<\frac 1 2$ . (In the constructions that follow the parameter $\tau'$ will eventually be chosen very close to $\tau$ .) Let $\epsilon'=\epsilon(\tau'-\tau)/(\tau'+\tau)$ , so that $0<\epsilon'<\epsilon$ . Consider the continuous piecewise linear function $h=h_{-\tau',\tau,\epsilon,\epsilon'}:(0,\infty)\to{\mathbb{R}}$ defined by the following conditions:

• – h coincides with $\ell_{-\tau}$ on $(0,1-\epsilon]$ ;

• – h is linear of slope $-\tau'$ on the interval $[1-\epsilon,1-\epsilon']$ ;

• – h is linear of slope $\tau$ on the interval $[1-\epsilon',1]$ ;

• – $h(1)=0$ ;

• – h coincides with $\ell_{-\tau}$ on the interval $[1,\infty)$ .

We call $h=h_{-\tau',\tau,\epsilon,\epsilon'}$ the $(-\tau',\tau,\epsilon,\epsilon')$ -dent on the function $\ell_{-\tau}$ . See Figure 4.

Figure 4.

Hamiltonian profile $h_{-\tau',\tau,\epsilon,\epsilon'}$ for Poincaré duality.

Figure 4 Long description

The diagram features a coordinate system with horizontal and vertical axes. The horizontal axis is labeled with variables such as 1/4, 1, and r, while the vertical axis is marked with values like 3τ/4, ετ, and -ε′τ. The diagram is divided into three regions labeled I, II, and III. Region I is located in the upper left quadrant, Region II in the lower left quadrant, and Region III in the lower right quadrant. Various lines and arrows indicate transitions and relationships between these regions, with labels such as -τ, -τ′, 1-ε, and 1-ε′. The diagram visually represents the Hamiltonian profile h-τ′,τ,ε,ε′ for Poincar duality, illustrating the interactions and boundaries between different regions.

Let $\tilde h$ be a smoothing of h with the following properties:

• – outside a small neighbourhood of $r=\frac 1 4$ and a small neighbourhood of the closed interval $[1-\epsilon,1]$ , the function $\tilde h$ coincides with h. In particular, $\tilde h$ is constant equal to $3\tau/4$ for $r\le \frac 1 4-\delta$ and is linear of slope $-\tau$ for $r\in[\frac 1 4+\delta,1-\epsilon-\delta]$ and $r\ge 1+\delta$ for some small $\delta\gt 0$ ;

• – inside the neighbourhood of the closed interval $[1-\epsilon,1]$ , and outside small neighbourhoods of $r=1-\epsilon$ , $r=1-\epsilon'$ , and $r=1$ , the function $\tilde h$ is linear of negative slope $-\tau'_1$ smaller but close to $-\tau'$ , then linear of positive slope $\tau_1$ larger but close to $\tau$ ;

• – inside the neighbourhood of $r=\frac 1 4$ where its derivative lies in $(-\tau,0)$ , the function $\tilde h$ is strictly concave;

• – inside the neighbourhood of $r=1-\epsilon$ where its derivative lies in $(-\tau'_1,-\tau)$ , the function $\tilde h$ is strictly concave;

• – in a neighbourhood of $r=1-\epsilon'$ the function $\tilde h$ is constant equal to $-\epsilon'\tau$ on some open interval, and is strictly convex as its slope varies in $(-\tau'_1,0)$ and $(0,\tau_1)$ ;

• – in a neighbourhood of $r=1$ the function $\tilde h$ is constant equal to 0 on some open interval, and is strictly concave as its slope varies in $(0,\tau_1)$ and $(-\tau,0)$ .

A function $\tilde h$ as above can be interpreted as a Hamiltonian $\tilde H:\widehat V\to{\mathbb{R}}$ by extending it as constant equal to $3\tau/4$ over $\widehat V\setminus (0,\infty)\times {\partial} V$ .

Assume now that $\tau$ does not belong to the action spectrum of ${\partial} V$ . Since the latter is a closed set in ${\mathbb{R}}$ , we can choose the other parameters $\tau',\tau_1,\tau'_1$ such that $\tau<\tau_1<\tau'<\tau'_1$ and such that the whole interval $[\tau,\tau'_1]$ does not intersect the spectrum. (In our later proof of Proposition 5.12 we will have to diminish even further the parameter $\tau'_1\gt \tau$ .) The parameter $\epsilon$ is chosen arbitrarily, while the parameter $\epsilon'$ is determined by $\epsilon$ , $\tau$ , and $\tau'$ , and is typically much smaller than $\epsilon$ . The 1-periodic orbits of $\tilde H$ fall then into three groups.

• – Group I consists of constants in $V_{\frac 1 4}$ and nonconstant orbits in the concavity region near $r=\frac 1 4$ .

• – Group II consists of orbits located in a neighbourhood of $r=1-\epsilon'$ , and these are themselves of three types: constants on the trivial cobordism which constitutes the minimal level (type $\mathrm{II}^0$ ), nonconstant orbits in the convexity region near its negative boundary (type $\mathrm{II}^-$ ), nonconstant orbits in the convexity region near its positive boundary (type $\mathrm{II}^+$ ).

• – Group III consists of orbits located in a neighbourhood of $r=1$ , and these are again of three types: constants on the trivial cobordism which constitutes the maximal level (type $\mathrm{III}^0$ ), nonconstant orbits in the concavity region of positive slope near its negative boundary (type $\mathrm{III}^-$ ), nonconstant orbits in the concavity region of negative slope near its positive boundary (type $\mathrm{III}^+$ ).

Let $H=H_\tau$ be a $C^2$ -small perturbation of $\tilde H$ , time-dependent and supported near the nonconstant periodic orbits in the concavity or convexity regions, time-independent Morse in the flat regions, and such that in each flat region the gradient of the Morse perturbation is pointing outwards along the boundary if the latter is adjacent to a convexity region, and is pointing inwards along the boundary if the latter is adjacent to a concavity region. The orbits of H naturally fall into classes I, II, and III as above, and their action is close to the action of the corresponding orbits of $\tilde H$ .

Let (a,b) be a fixed action window with $-\infty<a<0<b<\infty$ , and assume $\tau$ is large enough so that $-3\tau/4 < a$ . Then the action of all orbits in group I lies below the action window (a,b). Denote by

$$ (C_{a,b},{\partial}) := FC_*^{(a,b)}(H)$$

the Floer complex in the action window (a,b). Note that, up to canonical chain homotopy equivalence, this complex does not depend on $H=H_\tau$ as long as $\tau\gt \max\{4|a|/3,b\}$ . The generators of $C_{a,b}$ are orbits of types II and III, and we can write

$$ C_{a,b} = C_{a,b}^-\oplus C_{a,b}^+$$

where $C_{a,b}^-$ is the submodule generated by orbits of type III, and $C_{a,b}^+$ is the submodule generated by orbits of type II.

It follows from [Reference Cieliebak and OanceaCO18, Lemmas 2.2, 2.3, and 2.5] that $C_{a,b}^-$ is a subcomplex if the size of the perturbation $H-\tilde H$ is small enough. In particular we can think of $C_{a,b}$ as the cone C(f), where $f:C_{a,b}^+\to C_{a,b}^-[-1]$ is the part of the differential which maps elements of $C_{a,b}^+$ to elements of $C_{a,b}^-$ .

The Hamiltonian H is homotopic by a monotone homotopy supported in an open neighbourhood of the region $\{1-\epsilon\le r\le 1\}$ to the Hamiltonian $L=L_\tau$ which coincides with H in an open neighbourhood of $V_{\frac 1 4}$ and which is linear of slope $-\tau$ outside that neighbourhood. This homotopy defines a chain map $FC_*^{(a,b)}(H)\to FC_*^{(a,b)}(L)=0$ .

Choose now $\epsilon\gt 0$ small enough so that the Hamiltonians H and L are $C^0$ -close. By [Reference Cieliebak and OanceaCO18, Lemma 7.2], the reverse homotopy from L to H then induces a chain map $0=FC_*^{(a,b)}(L)\to FC_*^{(a,b)}(H)$ which is a homotopy inverse for the previous map. The homotopy of homotopies between the composition of these two homotopies and the constant homotopy for H induces a chain homotopy

$$ K:C_{a,b}\to C_{a,b}[1],$$

such that

$$ \mathrm{Id}_{C_{a,b}}= [{\partial},K].$$

It follows from the discussion in § 4.3 and from the definition of symplectic (co)homology in [Reference Cieliebak and OanceaCO18] that the map $f:C_{a,b}^+\to C_{a,b}^-[-1]$ above induces on homology the Poincaré duality isomorphism

$$ f_*:SH_*^{(a,b)}({\partial} V)\stackrel{\cong}\longrightarrow SH^{1-*}_{(-b,-a)}({\partial} V)$$

from symplectic homology to cohomology in truncated action. (This is also a consequence of [Reference Cieliebak and OanceaCO24a].) Consider now the pair-of-pants product

$$ C_{a,b}\otimes C_{a,b}\to FC_*^{(a+b,2b)}(2H),$$

as in § 4.2, where we use a 1-form $\beta$ satisfying $d\beta=0$ with positive weights 1 and negative weight 2. The Hamiltonian 2H is of the type $H_{2\tau}$ , $C^0$ -close to the corresponding linear Hamiltonian $L_{2\tau}$ . Thus $FC_*^{(a+b,2b)}(2H)$ can be identified with $C_{a+b,2b}$ and we obtain a degree $-n$ product

$$ \mu:C_{a,b}\otimes C_{a,b}\to C_{a+b,2b}.$$

Note that the target also splits as $C_{a+b,2b}=C_{a+b,2b}^-\oplus C_{a+b,2b}^+$ , with $C_{a+b,2b}^-$ a subcomplex, and in the notation of § 4.3 we have

again as a consequence of [Reference Cieliebak and OanceaCO18, Lemmas 2.2, 2.3, and 2.5]. Hence, we are in the situation of § 4.3 with $C=C_{a,b}$ and $\overline C=C_{a+b,2b}$ . In the notation of that section, it follows that $\mu^+=\mu^{++}_+$ descends to a product on homology

$$ \mu^+_*:SH_*^{(a,b)}({\partial} V)\otimes SH_*^{(a,b)}({\partial} V)\stackrel{[-n]}\longrightarrow SH_*^{(a+b,2b)}({\partial} V).$$

Moreover, Lemma 4.7 implies:

1. (a) the primary product $\mu^-=\mu^{--}_-$ vanishes on homology (cohomology after dual identification), i.e.

   $$ \mu^-_*=0:SH^{-*}_{(-b,-a)}({\partial} V)\otimes SH^{-*}_{(-b,-a)}({\partial} V)\to SH^{-*}_{(-2b,-a-b)}({\partial} V);$$

2. (b) the vanishing of the primary product in two ways gives rise to a secondary product

   $$ \sigma^-_*:SH^{1-*}_{(-b,-a)}({\partial} V)\otimes SH^{1-*}_{(-b,-a)}({\partial} V)\stackrel{[-n]}\longrightarrow SH^{1-*}_{(-2b,-a-b)}({\partial} V);$$

3. (c) the products $\mu^+_*$ and $\sigma^-_*$ on homology are related via the Poincaré duality isomorphism $f_*$ and its inverse $g_*$ as follows.

By construction, the maps in this diagram are compatible with the action filtrations, so for $a<a'$ and $b<b'$ the diagram is related to the corresponding diagram for a’,b’ via the action truncation maps

$$ SH^{(a,b)}_*({\partial} V)\to SH^{(a',b')}_*({\partial} V),\quad SH_{(-b,-a)}^{1-*}({\partial} V)\to SH_{(-b',-a')}^{1-*}({\partial} V).$$

Passing first to the inverse limit as $a\to-\infty$ and then to the direct limit as $b\to\infty$ , we obtain a degree $-n$ product $\boldsymbol{\mu}=\mu^+_*$ on $SH_*({\partial} V)$ and a degree $n-1$ secondary product $\boldsymbol{\lambda}^\vee\tau=\sigma^-_*$ on $SH^*({\partial} V)$ which are related via the Poincaré duality isomorphism $PD=f_*$ as follows.

This concludes the proof of Theorem 4.8.

4.5 Poincaré duality with coproducts

We keep the set-up of § 4.4, with V a Liouville domain of dimension 2n. The degree $-n$ product on $SH_*({\partial} V)$ can be rephrased as a degree n coproduct on $SH^*({\partial} V)$ , see § 4.2. On the other hand $SH_*({\partial} V)$ carries a coproduct of degree $-n$ determined by counts of rigid pairs-of-pants in combination with suitable action truncations. We refer to this coproduct as the primary coproduct on $SH_*({\partial} V)$ .