Loop coproduct in Morse and Floer homology

By a well-known theorem of Viterbo, the symplectic homology of the cotangent bundle of a closed manifold is isomorphic to the homology of its loop space. In this paper, we extend the scope of this isomorphism in several directions. First, we give a direct definition of Rabinowitz loop homology in terms of Morse theory on the loop space and prove that its product agrees with the pair-of-pants product on Rabinowitz Floer homology. The proof uses compactified moduli spaces of punctured annuli. Second, we prove that, when restricted to positive Floer homology, resp. loop space homology relative to the constant loops, the Viterbo isomorphism intertwines various constructions of secondary pair-of-pants coproducts with the loop homology coproduct. Third, we introduce reduced loop homology, which is a common domain of definition for a canonical reduction of the loop product and for extensions of the loop homology coproduct which together define the structure of a commutative cocommutative unital infinitesimal anti-symmetric bialgebra. Along the way, we show that the Abbondandolo–Schwarz quasi-isomorphism going from the Floer complex of quadratic Hamiltonians to the Morse complex of the energy functional can be turned into a filtered chain isomorphism using linear Hamiltonians and the square root of the energy functional.


Introduction
For a closed manifold M , there are canonical isomorphisms Here, we use coefficients in any commutative ring R, twisted in the first group by a suitable local system η which restricts to the orientation local system This article is part of the topical collection "Symplectic geometry-A Festschrift in honour of Claude Viterbo's 60th birthday" edited by Helmut Hofer, Alberto Abbondandolo, Urs Frauenfelder, and Felix Schlenk.
T * q M . See [16]. We do not spell out these results and focus on closed strings in this paper.
The first two isomorphisms in (1) are obtained by dividing out the constant loops, resp. the action zero part in the chain of isomorphisms (2) • The map ι intertwines the pair-of-pants products, and the map π intertwines the pair-of-pants coproducts. • The "almost splitting" i satisfies π • i = j and intertwines the product dual to λ F on SH 1− * >0 (D * M ) with the pair-of-pants product on SH * (S * M ). • The "almost splitting" p satisfies p•ι = q and intertwines the coproduct on SH * (S * M ) with the continuation coproduct λ F on SH >0 * (D * M ).
Here, the map e is multiplication with the Euler characteristic of M in degree 0. From this perspective, and up to some discrepancy at the constant loops, both the pair-of-pants product on SH * (D * M ) and the product dual to λ F on SH 1− * >0 (D * M ) appear as "components" of the pair-of-pants product on SH * (S * M ). See [21, §7].
One difficulty with the proof of Theorem 1.5 is the lack of an obvious chain map inducing the isomorphism SH * (S * M ) ∼ = H * Λ, due to the fact that the natural chain maps inducing Viterbo's isomorphisms on homology and cohomology go in opposite directions. We overcome this difficulty using the theory of A + 2 -structures from [20]. We will prove that the Abbondandolo-Schwarz map on chain level yields a quasi-isomorphism of A + 2 -structures, and then appeal to algebraic results from [20] concerning such structures and their associated cones.
Starting from the exact sequence (6), we define in this paper reduced loop homology and cohomology H * Λ = coker ε, H * Λ = ker ε. Each such extension λ defines together with the loop product μ the structure of a commutative cocommutative unital infinitesimal anti-symmetric bialgebra on H * Λ = H * +n Λ. In particular, the following relation holds: where we denote 1l the identity map and 1 the unit for the product μ.
We refer to [19,21] for the definition of a commutative cocommutative unital infinitesimal anti-symmetric bialgebra. The extensions of the coproduct depend on the choice of auxiliary data consisting of a Morse function on M with a unique maximum, a Morse-Smale gradient vector field, and a vector field with nondegenerate zeroes located away from the (n − 1)-skeleton. We discuss this dependence in Sect. 4. The coproduct is independent of all choices when H 1 M = 0 (Proposition 4.7), and in that case it also vanishes on the unit 1 (Corollary 4.9), so that the above relation becomes the unital infinitesimal relation λμ = (μ ⊗ 1l)(1l ⊗ λ) + (1 ⊗ 1l)(λ ⊗ 1l).
Structure of the paper. In Sect. 2, we define the notion of a special A + 2structure and prove that the Morse complex of the energy functional on loop space carries such a structure. In particular, this includes a Morse theoretic definition of the loop coproduct.
In Sect. 3 we construct a special A + 2 -structure on the chain complexes underlying symplectic homology of D * M .
In Sect. 4, we discuss extensions of the loop coproduct to reduced homology, and also the dependence of these extensions on choices.
In Sect. 5, we revisit the Viterbo isomorphism between symplectic homology of the cotangent bundle and loop space homology. We show that the Abbondandolo-Schwarz map Ψ : SH * (D * M ) → H * (Λ; η), which was originally constructed using asymptotically quadratic Hamiltonians and as such did not preserve the natural filtrations (at the source by the non-Hamiltonian action, and at the target by the square root of the energy), can be made to preserve these filtrations when implemented for the linear Hamiltonians used in the definition of symplectic homology. As such, Ψ becomes an isomorphism at chain level. This uses a length vs. action estimate inspired by [17].
In Sect. 6, we prove that the isomorphism Ψ intertwines the special A + 2structures of Sect. 2 and Sect. 3, which together with algebraic results from [20] yields Theorem 1.5. Our proof uses homotopies in certain compactified moduli spaces of punctured annuli. In Remark 6.2, we discuss some related open questions involving the two chain-level isomorphisms between Morse and Floer complexes constructed by Abbondandolo-Schwarz in [1,6].
In Sect. 7, we restrict to positive symplectic homology on the symplectic side, respectively, to loop homology rel constant loops on the topological side. We relate there the coproduct λ F resulting from Sect. 3 to the other secondary coproducts mentioned above, thus proving Theorem 1.1. In particular, this implies that the secondary coproduct defined by Abbondandolo and Schwarz in [4] corresponds under the isomorphism Ψ (restricted to the positive range) to the loop coproduct. For completeness, we also give a direct proof of this last fact in Sect. 7.4.
In Sect. 8, we compute the extended coproducts on reduced loop homology of odd-dimensional spheres S n . For n ≥ 3, these coproducts are canonical, but for n = 1 one sees explicitly the dependence on the choice of auxiliary data discussed in Sect. 4.
The Appendix contains a complete discussion of local systems on free loop spaces and their behaviour with respect to the loop product and coproduct. Local systems are unavoidable in the context of manifolds which are not orientable [7,31], and also in the context of the correspondence between symplectic homology of D * M and loop space homology of M [5,7,30]. They also proved useful in applications [9].

A + 2 -algebras
In this subsection, we recall from [20] the definition and basic properties of A + 2 -algebras. We will restrict to the case of special A + 2 -algebras which suffices for our purposes.
Let R be a commutative ring with unit, and (A, ∂) a differential graded R-module. Let A ∨ * = Hom R (A − * , R) be its graded dual, and ev : A ∨ ⊗A → R the canonical evaluation map. We denote Definition 2.1. A special A + 2 -structure on (A, ∂) consists of the following Rlinear maps: • the continuation quadratic vector c 0 : R → A ⊗ A, of degree 0; • the secondary continuation quadratic vector Q 0 : R → A ⊗ A, of degree 1; • the product μ : A ⊗ A → A, of degree 0; • the secondary coproduct λ : A → A ⊗ A, of degree 1.
The continuation quadratic vector c 0 gives rise to the continuation map O O Remark 2.5. The word "special" refers to the conditions (5) in Definition 2.1 and Θc = 0 in Definition 2.3. These conditions are imposed in order to simplify the algebra in [20]. These conditions, as well as the hypothesis n = 2 from Theorems 1.5 and 6.1, can be removed by upgrading the theory of A + 2 -structure to a theory of A + 3 -structures, which would include arity 3 operations.
Remark 2.6. (a) The conditions in Definition 2.1 imply (Proposition 2.2) that μ and λ induce a product μ on H * (Cone(c)). Associativity of μ requires further compatibilities between μ and λ, one of them being the "unital infinitesimal relation" [16,20,21]. (b) The conditions in Definition 2.1 imply that λ descends to the "reduced homology" H * (A/im c) and the map π in (6) intertwines it with a naturally defined coproduct λ on the cone; see [21] for further details.

A +
2 -structure on the Morse complex of the loop space Let now M be a closed connected manifold of dimension n. For simplicity, we assume that M is oriented and we use untwisted coefficients in a commutative ring R; the necessary adjustments with twisted coefficients are explained in Appendix A. We denote S 1 :=R/Z and Λ:=W 1,2 (S 1 , M).
Our goal in this subsection is to construct an A + 2 -structure on the Morse complex of Λ. The analysis underlying the Morse complex is identical to the one in [1,24] and we refer to there for details.
The Morse complex. Consider a smooth Lagrangian L : S 1 × T M → R which outside a compact set has the form L(t, q, v) = 1 2 |v| 2 − V ∞ (t, q) for a smooth potential V ∞ : S 1 × M → R. It induces an action functional which we can assume to be a Morse function. This functional is continuously differentiable and twice Gâteaux-differentiable on the space of loops of class W 1,2 , but in general it is not smooth (unless L is everywhere quadratic) [2]. Abbondandolo and Schwarz proved in [2] that it admits a negative pseudogradient vector field which is smooth and Morse-Smale. The latter condition means that for all a, b ∈ Crit(S L ) the unstable manifold W − (a) and the stable manifold W + (b) with respect to the negative pseudo-gradient intersect transversely in a manifold of dimension ind(a) − ind(b), where ind(a) denotes the Morse index with respect to S L . Let (MC * , ∂) be the Morse complex of S L with R-coefficients. It is graded by the Morse index and the differential is given by Then, ∂ • ∂ = 0 and its homology MH * is isomorphic to the singular homology H * Λ. We will assume in addition that near the zero section L(t, q, v) = We assume that L| M has a unique minimum q 0 and a unique maximum q Max . We denote by χ = χ(M ) the Euler characteristic of M and define the R-linear map c 0 : R → MC 0 ⊗ MC 0 by The element c 0 is clearly a cycle and we actually have τ c 0 = c 0 . Note, however, that the secondary continuation element Q 0 that we construct in the sequel may be nonzero. See also Sect. 4.  corresponding to broken pseudo-gradient lines. Therefore, we have i.e. μ satisfies condition (3)  The loop product is associative, and this is reflected at chain level by the fact that μ is associative up to chain homotopy. The critical point q Max is a cycle which is a two-sided unit for μ up to homotopy. Moreover, the subcomplex of constant loops MC =0 * ⊂ MC * is stable under μ and we can choose the Morse data such that q Max is a strict unit for the restriction of μ to MC =0 * . The coproduct λ. We fix a small vector field v on M with nondegenerate zeroes such that the only periodic orbits of v with period 1 are its zeroes. (The last property can be arranged, e.g. by choosing v gradient-like near its critical points; then the periods of nonconstant periodic orbits are uniformly bounded from below by a constant c > 0, so v/2c has the desired property.) Denote by the flow of v, i.e. the solution of the ordinary differential equation d dt f t = v•f t . It follows that the only fixed points of f = f 1 are the zeroes of v, each zero q is nondegenerate as a fixed point, and Remark 2.8. Alternatively, we could use the exponential map of some Riemannian metric to define a map M → M by q → exp q tv(q). Although this map differs from f t above, for v sufficiently small it shares its preceding properties and could be used in place of f t .

Consider now a generic family of vector fields
Note that, while v and −v have nondegenerate zeroes, this condition cannot be guaranteed for v τ . Genericity of the family means that the For each q ∈ M and τ ∈ [0, 1], we denote the induced path from q to f τ (q) by π τ q : [0, 1] → M, π τ q (t):=f τ t (q), and the inverse path by Recall from above the restriction and concatenation of paths. Now, for a, b, c ∈ Crit(S L ), we set Fig. 1.
Note that the matching conditions imply α(τ ) = f τ • α(0). This is a codimension n condition and, as the family v τ is generic, M 1 (a; b, c) is a transversely cut out manifold of dimension If its dimension equals zero this manifold is compact and defines a map If the dimension equals 1 it can be compactified to a compact 1-dimensional manifold with boundary Here, the first three terms correspond to broken pseudo-gradient lines and the last two terms to the intersection of M 1 (a; b, c) with the sets {τ = 1} and {τ = 0}, respectively. Therefore, we have where for i = 0, 1, we set Let us look more closely at the map λ 1 . For τ = 1 the matching conditions in M 1 (a; b, c) imply that α(0) = q is a fixed point of f 1 , the time-one flow of −v, and γ = q is the constant loop at q. Assuming that L| M has a unique minimum q 0 and the fixed points of f 1 are in general position with respect to the stable and unstable manifolds of L| M , the condition q ∈ W + (c) is only satisfied for c = q 0 . Thus, Choosing all fixed points of f 1 closely together, we can achieve that the terms on the right hand side corresponding to different q ∈ Fix(f 1 ) are in canonical bijection to each other. By the discussion before Remark 2.8, the terms corresponding to a fixed point q come with the sign ind −v (q). Since Since the last moduli space is the one in the definition of μ, we conclude or equivalently Similarly, we have or equivalently In conclusion, we obtain condition (4) in Definition 2.1, We define the secondary continuation quadratic vector Q 0 by Condition (2), i.e. τ c 0 −c 0 = [∂, Q 0 ], follows by inserting q Max into the relation for [∂, λ] and using that q Max is a strict two-sided unit for μ on MC =0 * . This is an instance of unital A + 2 -structure [20]. Note that Q 0 ∈ MC =0 * ⊗ MC =0 * for energy reasons. We now prove condition (5) in Definition 2.1. For n = 1 it holds because χ = 0, hence c = 0. We, therefore, assume w.l.o.g. n ≥ 2 and give the proof in two steps.
1. We first prove λc = 0. This follows from λ(q 0 ) = 0, which is seen as follows. The coefficient λ(q 0 ), x ⊗ y can only be nonzero if x, y are critical points of K. Since λ has degree 1 − n, we must have . We identify MC =0 * (L) with the Morse cochain complex MC n− * (V ). The cohomological index of Q 0 is 2n − 1, so its components must have degrees n − 1 and n. If n ≥ 2 these degrees are both positive, and therefore any component of Q 0 is killed by multiplication with q 0 because the latter has cohomological index n.
In summary, we have shown Proposition 2.9. Each vector field v on M satisfying the preceding conditions gives rise to a special A + 2 -structure (c 0 , Q 0 , μ, λ) on the Morse complex MC * of the functional S L : Λ → R. Remark 2.10. In the previous construction, we used an interpolating family of vector fields v τ such that v 1 = −v 0 . This choice is important because it ensures that the product on the Rabinowitz loop homology obtained from the A + 2 -structure via the cone construction is associative, and much more: in view of the isomorphism with the A + 2 -structure on symplectic homology proved in Sect. 6 and in view of [16,20], the resulting product fits into a graded Frobenius algebra structure on H * Λ.
While the construction of an A + 2 -structure would have worked with any choice of nondegenerate vector fields v 0 and v 1 at the endpoints of the parametrising interval, the necessity of the condition v 1 = −v 0 , which ensures these fine properties of the product, would become visible at chain level within a theory of A + 3 -structures. The development of such a theory is a matter for further study.
Remark 2.11. The description of M 1 τ =1 (a; b, q 0 ) and M 1 τ =0 (a; q 0 , c) above implies that λ 1 , λ 0 : MC * → (MC ⊗ MC) * −n are chain maps. By Eq. (9) they are chain homotopic, hence they induce the same "primary" coproduct Remark 2.12. Alternatively, we could define the loop coproduct using the spaces . Again the matching conditions imply α(τ ) = f τ • α(0), and M 1 (a; b, c) is a transversely cut out manifold of dimension ind(a) − ind(b) − ind(c) + 1 − n whose rigid counts define a map A discussion analogous to that for λ shows that Proposition 2.9 also holds with λ in place of λ. The obvious homotopies between the loops α τ 1 and α τ 2 in the definition of λ and the loops t in the definition of λ provide a special morphism between (c 0 , Q 0 , μ, λ) and (c 0 , Q 0 , μ, λ), where Q 0 = − λ(q Max ). We will use the restriction of the map λ to Morse chains modulo constants in the proof of Proposition 7.4.

A + 2 -structure for symplectic homology
As in the previous section, let M be a closed oriented manifold. We pick a Riemannian metric on M and denote by S * M ⊂ D * M ⊂ T * M its unit sphere resp. unit disc cotangent bundle. The latter is a Liouville domain whose completion is T * M . Its symplectic homology SH * (D * M ) is defined as the direct limit of the Floer homologies F H * (K) over Hamiltonians K : S 1 × T * M → R that are negative on D * M and linear outside a compact see; see [22] for general background on symplectic homology. The goal of this section is to construct a special A + 2 -structure on the chain complex underlying symplectic homology.

The continuation map c F
Recall from [22] that for Hamiltonians H K, we have a continuation map c H,K : F C * (H) → F C * (K), defined by counting Floer cylinders for an sdependent Hamiltonian H(s, ·) which agrees with K for small s, with H for large s, and which satisfies ∂ s H 0. In this subsection, we will describe the continuation map for a convex function k with k(0) = 0 and k(r) = μr for large r, with μ > 0 not in the length spectrum, and a potential V : M → R which has a unique maximum q 0 and a unique minimum q Max . For 1-periodic orbits x of −K and y of K, the coefficient c F x, y is given by the count of solutions u : R × S 1 → T * M of the Floer equation converging to x as s → +∞ and to y as s → −∞.
The Fredholm index of this problem is with equality iff x = y = q 0 for the maximum q 0 ∈ M of V . On the other hand, solutions u of (11) are in one-to-one correspondence to points x denotes the stable manifold of x with respect to ∇V . This shows that the Fredholm problem given by (10) resp. (11) is degenerate.
To perturb it, we denote by Skel k (V ) ⊂ M the k-skeleton, i.e. the union of the descending manifolds W − x of critical points of index k. We pick a 1-form η on M satisfying the following condition: All zeroes of η are nondegenerate and lie in M \ Skel n−1 (V ).
It gives rise to the flow generated by the vector field η on T * M , We pick a compactly supported function ρ : R → [0, ∞) with R ρ = 1 and perturb Eqs. (10) and (11) to and To understand solutions of the perturbed Morse equation, we choose φ, ρ such that φ ≡ 0 on [−1, 1] and supp(ρ) ⊂ [−1, 1]. Then, solutions u of (14) are in one-to-one correspondence to points , where the intersection is taken in T * M . By condition (12) this intersection is empty unless x = y = q 0 , in which case intersection points correspond to zeroes of η and their signed count equals the Euler characteristic χ of M . This shows that the only nontrivial term in the continuation map c F : and the corresponding quadratic vector c F 0 is given by c F 0 (1) = χq 0 ⊗ q 0 . In particular, c F 0 satisfies the closedness condition in Definition 2.1, and it is also symmetric τ c F 0 = c F 0 . Note that this holds without any symmetry assumptions on the data such as φ(−s) = −φ(s) or ρ(−s) = ρ(s). Note also that, although the definition of c F 0 on the chain level requires the choice of a pair (V, η) consisting of a Morse function V : M → R and a 1-form η on M subject to condition (12), the result does not depend on this choice. In contrast, the secondary continuation quadratic vector Q F 0 which we construct below may depend on this choice. See also Sect. 4.

The product μ F and coproduct λ F
The pair-of-pants product μ F : F C * (K) ⊗ F C * (K) → F C * (2K) (of degree −n) counts maps from a pair-of-pants satisfying a Floer equation with weights 1 at the two positive punctures and weight 2 at the negative puncture. The definition is entirely analogous to the one for the coproduct λ F given below, without the additional parameter τ . It is well-known that μ F is a chain map which is associative and graded commutative up to chain homotopy (see e.g. [3]), so condition (2) in Definition 2.1 holds.
The critical point q Max is a constant orbit and is a cycle which is a twosided unit for μ F up to homotopy. The subcomplex F C =0 * (K) ⊂ F C * (K) generated by small action orbits is stable under μ F and we can choose the auxiliary data such that q Max is a strict unit for the restriction of μ F to F C =0 * (K).
In [21], a secondary coproduct λ is defined in terms of continuation maps on the reduced symplectic homology of a large class of Weinstein domains which includes cotangent bundles. See also [16,20]. In this subsection, we recall its definition for D * M ; we will call it the continuation coproduct and denote it by λ F .
As before, we denote by r = |p| the radial coordinate on T * M . Let K = K μ be a convex smoothing of the Hamiltonian which is zero on D * M and equals r → μr outside D * M . Then, 2K = K 2μ and −K = K −μ are the corresponding Hamiltonians of slopes 2μ and −μ, respectively.
Let Σ be the 3-punctured Riemann sphere, where we view one puncture as positive (input) and the other two as negative (outputs). We fix cylindrical coordinates (s, t) ∈ [0, ∞) × S 1 near the positive puncture and (s, t) ∈ (−∞, 0] × S 1 near the negative punctures. Consider a 1-form β on Σ which equals B dt near the positive puncture and A i dt near the i-th negative puncture (i = 1, 2) for some A i , B ∈ R. We say that β has weights B, A 1 , A 2 . We, moreover, require dβ 0, which is possible iff We consider maps u : Σ → T * M satisfying the perturbed Cauchy-Riemann equation Near the punctures this becomes the Floer equation for the Hamiltonians BK and A i K, respectively, and the algebraic count of such maps defines a (primary) coproduct which has degree −n and decreases the Hamiltonian action.
The algebraic count of such pairs defines a (secondary) coproduct which has degree 1 − n and decreases the Hamiltonian action.
Let us analyse the contributions from τ = 0, 1. The algebraic count of cylinders with weights −1, 2 defines the continuation map (of degree 0) As explained in the previous subsection, to define c F , we perturb K by a Morse function V : M → R with a unique maximum at q 0 and a unique minimum at q Max . Moreover, we choose a family of 1-forms η τ on M such that η 0 = η 1 satisfies condition (12) (for all practical purposes one can think of η τ as being constant). Finally, we choose a family of compactly supported 1-forms α τ on Σ which for τ = 0, 1 agree with ρ(s)ds supported in the split off cylinder, for a function ρ : R → [0, ∞) satisfying R ρ = 1. For example, we can take α τ = (1 − τ )α 0 + τ α 1 where α 0 = ρ(s)ds supported on the first negative end, and α 1 = ρ(s)ds supported on the second negative end (below the level where the splitting happens at τ = 0, 1). With η τ the vector field on T * M corresponding to η τ , we replace the Cauchy-Riemann equation in the definition of λ F by With these choices, it follows from the discussion in the previous subsection that the only nontrivial terms in the continuation maps at τ = 0, 1 are c F q 0 = χq 0 , where χ is the Euler characteristic of M . As shown in Fig. 2, the contribution at τ = 0 consists of a pair-of-pants with one positive puncture of weight 1 and two negative punctures of weights −1 and 2, with a cylinder of weights −1 and 2 attached at the first negative puncture. We reinterpret this as a pair-of-pants with two positive punctures of weights 1 and one negative puncture of weight 2, with a cylinder with two negative punctures of weights 2 and 1 attached at the first positive puncture.
The preceding discussion shows that the count of these configurations corresponds to the composition (1 ⊗ μ F )(τ c F 0 ⊗ 1). A similar discussion at τ = 1 establishes that λ F satisfies condition (4) in Definition 2.1.
We define the secondary continuation quadratic vector Q F 0 by , follows by inserting q Max into the relation for [∂, λ F ] and using that q Max is a strict two-sided unit for μ on F C =0 * (K). This is an instance of unital A + 2 -structure [20]. Note that An inspection of the definition shows that Q F 0 coincides with the secondary continuation element defined in Sect. 4.2 by interpolating between the perturbing 1-form η and its opposite −η. See also [21].
It remains to prove condition (5) in Definition 2.1. For n = 1 it holds because χ = 0, so that c F 0 = 0. We, therefore, assume w.l.o.g. n ≥ 2 and, as in the Morse case, we prove condition (5) in two steps.
1. We first prove λ F c F = 0. This follows from λ F (q 0 ) = 0, which is seen as follows. For action reasons, the coefficient λ F (q 0 ), x ⊗ y can only be nonzero if x, y are critical points of K. Since λ F has degree 1 − n, we must . We identify the Floer subcomplex F C =0 * (K) generated by orbits of small action with the Morse cochain complex MC n− * (V ). The cohomological index of Q F 0 is 2n − 1, so its components must have degrees n − 1 and n. If n ≥ 2 these degrees are both positive, and therefore any component of Q 0 is killed by multiplication with q 0 because the latter has cohomological index n. In summary, we have shown Proposition 3.1. The operations c F 0 , Q F 0 , μ F , λ F on the Floer chain complexes F C * (K) resp. F C * (2K) satisfy the relations of a special A + 2 -structure. The operations c F 0 , Q F 0 μ F , λ F are compatible with Floer continuation maps between different Hamiltonians H K. We will refer to this structure as being the special A + 2 -structure for symplectic homology SH * (D * M ). Remark 3.2. In the previous construction, we imposed the condition η 0 = η 1 at the endpoints of the family of 1-forms η τ for the same reason why we imposed v 1 = −v 0 in the Morse case: this ensures that the product on Rabinowitz Floer homology obtained from the A + 2 -structure via the cone construction coincides with the product from [16] and fits into a graded Frobenius algebra structure on SH * (S * M ).
The construction of an A + 2 -structure would have worked with any choice of interpolating family η τ such that η 0 and η 1 satisfy (12). The necessity of the condition η 1 = η 0 for this fine behaviour of the product would become visible at chain level within a theory of A + 3 -structures.

Reduced loop homology
This section expands material from [21, §4] in the particular case of cotangent bundles. We assume that M is connected and orientable, and we work either with constant coefficients on the loop space, or with local coefficients η obtained by transgressing the 2nd Stiefel-Whitney class. In each of these two cases we have a commutative diagram where the vertical maps are restriction to, resp. inclusion of constant loops, and ε 0 is induced by multiplication with the Euler characteristic χ. From now on, we omit from the notation the local system η.
Definition 4.1. We define reduced loop homology, resp. cohomology, In the sequel, we restrict the discussion to reduced homology. Reduced cohomology features similar properties, with the roles of the product and coproduct being exchanged (as yet another instantiation of Poincaré duality for loop spaces [16]).
The behaviour of reduced homology with respect to the product is very robust. The image of ε is an ideal in H * (Λ) (see for example [38] or [16]), and therefore the loop product canonically descends to reduced homology H * (Λ).
In contrast, the behaviour of reduced homology with respect to the coproduct is very subtle. To describe it, the following variant of reduced loop homology arises naturally. A straightforward calculation shows that we have a canonical isomorphism 8. This is the case if M is orientable and if we use a local system that is constant on the component of contractible loops, or if χ = 0, or if R is 2-torsion. We place ourselves from now on in this setup, so that we do not need to distinguish between H * (Λ) and H * (Λ, χ · point).
The loop coproduct is canonically defined on H * (Λ, Λ 0 ). We now explain that it always extends to H * (Λ, χ · point) (and hence to H * (Λ) under our assumptions). However, this extension is not canonical. The extension depends on a choice of vector field with nondegenerate zeroes and on the choice of a Morse function on M . We will completely describe the dependence of the extension on the choice of vector field, and give sufficient conditions for independence of the extension on the choice of Morse function.

Reduced symplectic homology
We work with symplectic homology of D * M , our favourite model for loop space homology. Recalling notation from Sect. 3.1, we fix the following continuation data: • a Morse function V : M → R with a unique maximum q 0 . • a 1-form η on M which satisfies condition (12), i.e. the zeroes of η are nondegenerate and lie outside of Skel n−1 (V ) (this is equivalent to a vector field v on M whose zeroes have the same property).
We consider Hamiltonians K : T * M → R of the form K(q, p) = k(|p|) + V (q), where k(0) = 0 and k = k(r) is a linear function of r outside a compact set, of positive slope not belonging to the length spectrum. This data determines via Eq. (13) the Floer continuation map which has the property that the only generator on which it may be nonzero is q 0 . Moreover, we have computed in Sect. 3.1 that The continuation map can be equivalently interpreted as a quadratic vector . We emphasise that the chain-level expression of the continuation map is the same for any choice of continuation data (V, η).

Definition 4.3. The reduced Floer complex of K is
Its homology is the reduced Floer homology of K, denoted F H * (K).
The reduced symplectic homology SH * (D * M ) is the direct limit of reduced Floer homologies F H * (K) over Hamiltonians K which vanish on D * M and are linear outside a compact set, perturbed to have the form k(|p|)+V (q) near the zero section as above. 2 The relation . These maps are compatible with the continuation maps obtained by increasing the slope of K, giving rise in the limit (with field coefficients) to a well-defined coproduct of degree −n + 1, denoted A straightforward enhancement of the Viterbo-Abbondandolo-Schwarz isomorphism shows that the map Ψ induces an isomorphism between reduced homologies In particular, associated to a choice of continuation data (V, η) is a coproduct on H * (Λ). The key to understanding the dependence of the coproduct on the choice of continuation data (V, η) is the secondary continuation map, which we describe next.

The secondary continuation map
Homotopies between different choices of pairs (V, η) give rise to secondary operations which we describe in this subsection.
Consider two pairs (V i , η i ), i = 0, 1, satisfying the conditions of the previous subsection, i.e. V i : M → R is a Morse function with a unique maximum q i and η i a 1-form on M such that condition (12) holds. For i = 0, 1 let K i : T * M → R be associated Hamiltonians as in the previous subsection. After shifting V 0 , V 1 by constants we may assume without loss of generality As in the previous subsection, we pick a function φ : , be a smooth family of s-dependent Hamiltonians with the following properties: for |s| σ and σ 1.
Let η σ , σ ∈ [0, ∞), be a smooth family of 1-forms with η σ = η 1 for all σ 1. We consider pairs (σ, u) with σ ∈ [0, ∞) and u : R × S 1 → T * M solving the Floer equation and converging to 1-periodic orbits of ∓K 0 as s → ±∞. Their algebraic count gives rise to a degree 1 map with the Floer continuation maps c i : . The map Q factors through the action zero part which we will denote by Q =0 . Since the V i have unique maxima q i , it follows from the previous subsection that the only nontrivial contribution to c 0 is c 0 (q 0 ) = χq 0 . Similarly, the only nontrivial contribution to the composition c 10 c 1 c 01 sends q 0 → q 1 → χq 1 → χq 0 . This shows that the right hand side of Eq. (16) vanishes, and therefore Q descends to a map on homology which factors through the action zero part . For degree reasons nontrivial contributions can only occur for * = 0 and * = −1 and give maps In particular, we have shown

Dependence of the continuation coproduct on choices
In this subsection, we discuss the dependence of the continuation coproduct λ F on the data (V, η) of a Morse function and a 1-form on M .
We consider the setup of the previous subsection and retain the terminology from there. Thus, we are given two pairs ( be the degree 2−n map defined by the 2-parametric family of Floer problems depicted in Fig. 3.
These Floer problems are defined in terms of a 2-parametric family of Hamiltonian valued 1-forms on the 3-punctured sphere with asymptotics and degenerations as in the figure, and a 2-parametric family of 1-forms on M which agree with η 1,τ , τ ∈ [0, 1] on the top side, with η 1 on the two top slanted sides, and with η 0,τ , τ ∈ [0, 1] on the bottom side of the hexagon.
On the reduced Floer chain complex the compositions along the top vertical sides vanish because they factor through the continuation map c 1 : are the degree −n operations appearing at the ends of the continuation coproduct λ F 0 as in Fig. 2. All the maps appearing on the right hand side of the last displayed equation are chain maps, so they descend to maps on reduced Floer homology (denoted by the same letters) satisfying Passing to the direct limit over Hamiltonians K 0 , K 1 as above, we have therefore shown where Q is the secondary continuation map of the previous subsection and λ F 00 , λ F 01 are induced by the maps defined above. Remark 4.6. The right hand side of the previous equation can be rephrased in terms of the secondary continuation map and the product μ F . We refer to [21, §4.3] for further details.  Remark 4.8. In Sect. 3.2, we defined the coproduct λ F using a family η τ , τ ∈ [0, 1] with equal endpoints η 0 = η 1 = η. The proof of Proposition 4.5 shows that, under the assumption H 1 (M ) = 0, the coproduct can be defined using families η τ with arbitrary endpoints satisfying condition (12) (in particular, we can take η 1 = −η 0 ). This observation simplifies the computations in Sect. 8 for spheres of dimension > 1 by allowing the use of constant families of vector fields v τ ≡ v for the topological definition of the coproduct. A proof of this result in a more general setting is given in [21, §4], based on the vanishing of the secondary continuation map. We give here a topological proof, see also Sect. 8.2 for the case of spheres of odd dimension ≥ 3.
Proof. We work on the topological side H * Λ and compute, as in Sect. 8.2, the image of the fundamental class 1 by representing it by constant loops and using a constant vector field v τ ≡ v with isolated nondegenerate zeroes.
If v has no zeroes then its image under the coproduct is zero because it is represented by the empty chain. In the general case, the image is a degenerate 1-chain, hence vanishes in homology. is isomorphic to the loop homology H * (Λ) (Viterbo [39], Abbondandolo-Schwarz [1,5], Salamon-Weber [35], Abouzaid [7]). Here, we use coefficients twisted by the local system σ defined by transgressing the second Stiefel-Whitney class, cf. Appendix A. We drop the local system σ from the notation in the rest of this section.

Viterbo's isomorphism revisited
The construction most relevant for our purposes is the chain map from the Floer complex of a Hamiltonian H : S 1 × T * M → R to the Morse complex of an action functional S : Λ → R on the loop space defined in [4]. When applied to a fibrewise quadratic Hamiltonian H and the action functional S L associated to its Legendre transform L, it induces an isomorphism on homology intertwining the pair-of-pants product with the loop product [4].
One annoying feature of the map Ψ has been that, in contrast to its chain homotopy inverse Φ : MC * (S L ) → F C * (H), it does not preserve the action filtrations. This would make it unsuitable for some of our applications in [16] such as those concerned with critical values. Using an estimate inspired by [17], we show in this section that Ψ does preserve suitable action filtrations when applied to fibrewise linear Hamiltonians rather than fibrewise quadratic ones.

Floer homology
Consider a smooth time-periodic Hamiltonian H : S 1 × T * M → R which outside a compact set is either fibrewise quadratic, or linear with slope not in the action spectrum. It induces a smooth Hamiltonian action functional Its critical points are 1-periodic orbits x, which we can assume to be nondegenerate with Conley-Zehnder index CZ(x). Let J be a compatible almost complex structure on T * M and denote the Cauchy-Riemann operator with Hamiltonian perturbation on u : Let F C * (H) be the free R-module generated by Crit(A H ) and graded by the Conley-Zehnder index. The Floer differential is given by where #M(x; y) denotes the signed count of points in the oriented 0-dimensional manifold Then, ∂ F • ∂ F = 0 and its homology F H * (H) is isomorphic to the symplectic homology SH * (T * M ) if H is quadratic. If H is linear, we obtain an isomorphism to SH * (T * M ) in the direct limit as the slope goes to infinity.

The isomorphism Φ
Suppose now that H is fibrewise convex with fibrewise Legendre transform L : Following [1], for a ∈ Crit(S L ) and x ∈ Crit(A H ), we consider the space It was shown in [1] that the induced map on homology is an isomorphism The definition of the Legendre transform yields the estimate

It follows that
whenever M(a; x) is nonempty, so Φ decreases action.

The isomorphism Ψ
Consider once again a fibrewise quadratic Hamiltonian H : Following [4,17], for x ∈ Crit(A H ) and a ∈ Crit(S L ), we define where W + (a) is the stable manifold of a for the negative pseudo-gradient flow of S L , see Fig. 4.
The signed count of 0-dimensional spaces M(x; a) defines a chain map The induced map on homology is an isomorphism which is the inverse of Φ * and intertwines the pair-of-pants product with the loop product. This was shown by Abbondandolo and Schwarz [4] with Z/2-coefficients, and by Abouzaid [7] (following work of Kragh [30], see also Abbondandolo-Schwarz [5]) with general coefficients, twisted on H * Λ by a suitable local system, see Appendix A. Moreover, Abouzaid proved that Ψ * is an isomorphism of twisted BV algebras. Unfortunately, the map Ψ does not preserve the action filtrations. This already happens for a classical Hamiltonian H(q, p) so the middle inequality goes in the wrong direction (even if V = 0).

An action estimate for Floer half-cylinders
Now, we will replace the quadratic Hamiltonians from the previous subsections by Hamiltonians of the shape used in the definition of symplectic homology. For Floer half-cylinders of such Hamiltonians, we will estimate the length of their boundary loop on the zero section by the Hamiltonian action at +∞. We equip M with a Riemannian metric and choose the following data. The Riemannian metric on M induces a canonical almost complex structure J st on T * M compatible with the symplectic form ω st = dp ∧ dq (Nagano [32], Tachibana-Okumura [37], see also [11,Ch. 9]). In geodesic normal coordinates q i at a point q and dual coordinates p i it is given by We pick a nondecreasing smooth function ρ : [0, ∞) → (0, ∞) with ρ(r) ≡ 1 near r = 0 and ρ(r) = r for large r. Then, (in geodesic normal coordinates) defines a compatible almost complex structure on T * M which agrees with J st near the zero section and is cylindrical outside the unit cotangent bundle.
We view r(q, p) = |p| as a function on T * M . Then, on T * M \M , we have

Consider a Hamiltonian of the form
Then, its Hamiltonian vector field equals X H = h (r)R, where R is the Reeb vector field of (S * M, α). The symplectic and Hamiltonian actions of a nonconstant 1-periodic Hamiltonian orbit x : S 1 → T * M are given by Given a slope μ > 0 which is not in the action spectrum of (S * M, α) and any ε > 0, we can pick h with the following properties: Specifically, we choose 0 < δ ≤ ε/μ, we consider a smooth function β : β . This expression differentiates to μ(r − 1)β ≥ 0 and vanishes on [0, 1], hence it is nonnegative for r ≥ 0. On the other hand, we have an upper bound μ (r − 1)β − r 1 β ≤ μδ for r ∈ (1, 1 + δ), and indeed for r ≥ 0. Given our choice δ ≤ ε/μ, this establishes the inequalities rh These inequalities imply that for each nonconstant 1-periodic Hamiltonian orbit x, we have With this choice of J and H, consider now as in the previous subsection a map u : Set q(t):=u(0, t) and denote its length by The following proposition is a special case of [17,Lemma 7.2]. Since the proof was only sketched there, we give a detailed proof below.

The first inequality is an equality if and only if u is contained in the halfcylinder over a closed geodesic q, in particular x is the lift of the geodesic q.
The idea of the proof is to show that Since the image u (0, ∞) × S 1 can hit the zero section M where α is undefined, the quantity (0,∞)×S 1 u * dα has to be interpreted as an improper integral as follows. Given ε > 0, let τ = τ ε : [0, ∞) → [0, ∞) be a smooth function with τ (r) 0 for all r, τ (r) = 0 near r = 0, and τ (r) = 1 for r ε, and consider the globally defined 1-form on T * M given by We now define The proof of Proposition 5.1 is based on the following lemma. Proof. In geodesic normal coordinates, we compute

For a vector of the form
At points where τ > 0, equality only holds for a = 0, and at points where τ = 0 and τ > 0 equality holds iff a is a multiple of p. Similarly, for a general Proof of Proposition 5.1. The proof consists in 3 steps.
Step 1. We prove that (0,∞)×S 1 u * dα ≥ 0. In view of Definition (19), it is enough to show that u * dα ε 0 on all of (0, ∞) × S 1 . To see this, recall that u satisfies the equation which is nonnegative by Lemma 5.2.
Step 2. Denote u σ = u(σ, ·) for σ > 0. We have To see this, we consider the map and denote as aboveq σ =q(σ, ·) for σ > 0. Since J = J st near the zero section, the maps u andq agree with their first derivatives along the boundary loop q at s = 0, hence u σ andq σ are C 1 -close for σ close to 0. On the other hand α ε is C 0 -bounded near the zero section uniformly with respect to ε → 0. These two facts imply that the integrals S 1 u * σ α ε and S 1q * σ α ε are C 0 -close for σ close to 0, uniformly with respect to ε → 0, and therefore We now prove that for all σ > 0, which implies the desired conclusion. Fix therefore σ > 0. Let On the one hand, we have We can, therefore, estimate Here, m(I ε ) is the measure of I ε , uniformly bounded by the length of the circle, C > 0 is a C 0 -bound on α ε near the 0-section, uniform with respect to ε → 0, and ε/σ is by definition the bound on |q(t)| on I ε . The estimate follows fromq σ = (q, σq) and the fact that the 1-form α ε only acts on the first component of the vectorq σ . Since lim ε 0 (q| S 1 \Iε ) = (q| S 1 \I0 ) = (q), equality (20) follows.

Conclusion. Combining
Step 3 with Step 1, we obtain the first inequality (q) x α in Proposition 5.1. Moreover, Lemma 5.2 (in the limit ε → 0) shows that this inequality is an equality if and only if u is contained in the half-cylinder over a closed geodesic.
The second inequality x α A H (x) follows from (18).

The isomorphism Ψ from symplectic to loop homology
Now, we adjust the definition of Ψ to symplectic homology. For J, H as in the previous subsection and x ∈ Crit(A H ), we define as before By Proposition 5.1, the loop q = u(0, ·) satisfies (q) A H (x). Moreover, the loop q is smooth and in particular has Sobolev class H 1 , hence following Anosov [10] it has a unique H 1 -reparametrisation q : S 1 → M , with |q| ≡ const and q(0) = q(0) (we say that q is parametrized proportionally to arclength, or PPAL). We have The energy defines a smooth Morse-Bott function on the loop space whose critical points are constant loops and geodesics parametrized proportionally to arclength. We denote by W ± (a) the unstable/stable manifolds of a ∈ Crit(E) with respect to ∇E. Now, for x ∈ Crit(A H ) and a ∈ Crit(E), we define An element u in this moduli space still looks as in Fig. 4, where now the loop q = u(0, ·) is reparametrized proportionally to arclength and then flown into a using the flow of −∇E. By Proposition 5.1, for u ∈ M(x; a), we have the estimate To define the map Ψ, we now perturb H and E by small Morse functions near the constant loops on M and the closed geodesics, and we generically perturb the almost complex structure J from the previous subsection. For The signed count of 0-dimensional spaces M(x; a) defines a chain map Here, MC * (E 1/2 ) denotes the Morse chain complex of E : Λ → R, graded by the Morse indices of E, but filtered by the square root E 1/2 (which is decreasing under the negative gradient flow of E). The action estimate (21) continues to hold for the perturbed data up to an arbitrarily small error, which we can make smaller than the smallest difference between lengths of geodesics below a given length μ. Thus, Ψ preserves the filtrations

The induced maps on filtered Floer homology
have upper triangular form with respect to the filtrations with ±1 on the diagonal (given by the half-cylinders over closed geodesics in Proposition 5.1), so they are isomorphisms. It follows from [4,7] that Ψ * intertwines the pairof-pants product with the loop product, as well as the corresponding BV operators. Passing to the direct limit over Hamiltonians H, we have thus proved Theorem 5.3. The map Ψ induces isomorphisms on filtered symplectic homology where the left hand side is filtered by non-Hamiltonian action and the right hand side by the square root of the energy. These isomorphisms intertwine the pair-of-pants product with the Chas-Sullivan loop product, as well as the corresponding BV operators.

Viterbo's isomorphism intertwines A + 2 -structures
We keep the setup from the previous section, so M is a closed oriented Riemannian manifold and D * M ⊂ T * M its unit disc cotangent bundle. In this section, we prove Theorem 1.5, which will be an immediate consequence of earlier results and the following theorem. , defined for an asymptotically quadratic Hamiltonian H. It would be interesting to clarify whether Φ also defines a morphism of special A + 2 -algebras. Abbondandolo-Schwarz constructed in [6] an action-preserving chainlevel isomorphism Ψ AS : F C * (H) → MC * (S L ) which is a chain homotopy inverse of Φ. They also argued that, from the perspective of the Legendre transform, the moduli spaces that define Ψ AS arise naturally from the moduli spaces for Φ. We expect that Ψ AS and our morphism Ψ can be connected by a suitable chain homotopy (we know that they induce the same map Φ −1 * in homology). It would also be interesting to clarify whether Ψ AS is a morphism of special A + 2 -algebras. We expect this to hold or fail for both Ψ AS and Φ simultaneously.
One can further ask whether Ψ AS and Ψ are homotopic as morphisms of A + 2 -structures. This would require in particular to develop the discussion of A + 2 -structures from [20] by defining such a notion of homotopy. To prove Theorem 6.1, we need to verify the conditions in Definition 2.3 for each chain map Ψ : F C * (K) → MC * (E 1/2 ) associated to a Hamiltonian K = K μ as in the previous subsection. The first part of condition (i) holds because Ψq 0 = q 0 , which follows directly from the definition of Ψ. Moreover, seen through the canonical identifications F C =0 , the restriction of Ψ to the energy zero Floer subcomplex acts as the identity. This shows that the second part of condition (i) is also satisfied.
The map Γ : F C * (K) ⊗ F C * (K) → MC * (E 1/2 ) in condition (ii) is defined by the count of elements in 0-dimensional moduli spaces of solutions to a 1-parametric mixed Floer-Morse problem which we describe below. Inspection of the boundary of the 1-dimensional moduli spaces of solutions shows that Γ satisfies condition (ii). This fact was previously proved in [4], which contains the description of an essentially equivalent map Γ.
The 1-parametric Floer-Morse problem is a count of Floer discs in T * M with two positive punctures and boundary on the zero section, followed by a Morse pseudo-gradient line in the loop space of M . It is obtained as a concatenation of 3 distinct 1-parametric Floer-Morse problems described by Fig. 5. On the first interval of parametrisation the underlying moduli space of curves is that of discs with 2 interior punctures and one boundary marked In this configuration the interior punctures, the node and the marked point are all aligned. On the second interval of parametrisation, we allow the marked point to move clockwise towards the node. At the positive end of this interval the marked point collides with the node and forms a disc bubble. However, this disc bubble is constant because the 0-section is an exact Lagrangian, so that we directly replace the configuration by one where the marked point lies at the node. On the third and last interval of parametrisation, we insert length T > 0 pseudo-gradient lines before imposing the incidence condition at the marked point. The positive end of this interval of parametrisation corresponds to T = ∞ and gives rise to the term μ(Ψ ⊗ Ψ) in the expression of [∂, Γ].
In Fig. 5, the dashed lines represent pseudo-gradient flow lines for the energy functional on loop space. We only represent them in the last two configurations depicted in Fig. 5 in order not to burden excessively the drawing. However, the reader should be aware that such pseudo-gradient lines are also present in the first five configurations from Fig. 5.
For further reference, it is convenient to write where Γ i , i = 1, 2, 3 corresponds to the count of elements in the 0-dimensional moduli spaces of solutions to the Floer-Morse problem restricted to the i-th interval of parametrisation for Γ. The remainder of this section is devoted to the proof of condition (iii). For this, we need to construct a chain homotopy The map Θ will be defined by a count of Floer maps to T * M defined over a 2-parametric family of punctured annuli. In the first subsection, we describe the underlying moduli space of conformal annuli.

Conformal annuli
A (conformal) annulus is a compact genus zero Riemann surface with two boundary components. By the uniformisation theorem (see for example [12]), each annulus is biholomorphic to [0, R] × R/Z with its standard complex structure for a unique R > 0 called its (conformal) modulus. The exponential map s + it → e 2π(s+it) sends the standard annulus onto the annulus It will be useful to consider slightly more general annuli in the Riemann sphere S 2 = C ∪ {∞}. A circle in S 2 is the transverse intersection of S 2 ⊂ R 3 with a plane. We will call a disc in S 2 an open domain D ⊂ S 2 bounded by a circle, and an annulus in S 2 a set D \ D for two discs D, D ⊂ S 2 satisfying D ⊂ D (with the induced complex structure).

Lemma 6.3. Every annulus A in S 2 of conformal modulus R can be mapped by a Möbius transformation onto the standard annulus
After applying a Möbius transformation, we may assume that D is the disc {z ∈ C | |z| < e 2πR }. Let D 1 ⊂ D be the unit disc. There exists a Möbius transformation φ of D sending a point z ∈ ∂D to a point z 1 ∈ ∂D 1 and the positive tangent direction to ∂D at z to the positive tangent direction to ∂D 1 at z 1 . Thus, φ sends ∂D to a circle tangent to ∂D 1 at z 1 , and since the annuli D\D and D\D 1 both have modulus R, we must have φ(∂D ) = ∂D 1 , hence φ(D ) = D 1 .
For each R, the standard annulus [0, R] × R/Z carries two canonical foliations: one by the line segments [0, R]×pt and one by the circles pt×R/Z. Moreover, these two foliations are invariant under the automorphism group of the annulus. Hence, by Lemma 6.3 each annulus in S 2 also carries two canonical foliations, one by circle segments connecting the two boundary components and one by circles, such that the foliations are orthogonal and the second one contains the two boundary loops. These two foliations can be intrinsically described as follows: the automorphism group of an annulus A is Aut(A) S 1 . The second foliation consists of the orbits of the S 1 -action. The first foliation is the unique foliation orthogonal to the first one. Its leaves connect the two boundary components because this is the case for a standard annulus. Figure 6 shows a 1-parametric family of annuli in C whose conformal moduli tend to 0 together with their canonical foliations. The domain at modulus 0 is the difference of two discs touching at one point, the node. Putting the node at the origin, the inversion z → 1/z maps this domain onto a horizontal strip in C (with the node at ∞) with its standard foliations by Opening up the node, we can conformally map it onto the standard disc with two boundary points corresponding to the node (since the map is not a Möbius transformation, the two foliations will not be by circle segments).
Annuli with aligned marked points. The relevant domains for our purposes are annuli with 3 marked points, one interior and one on each boundary component. We require that the 3 points are aligned, by which we mean that they lie on the same leaf of the canonical foliation connecting the two boundary components. (In the next subsection, the interior marked point will correspond to the input from the Floer complex and the boundary marked points will be the initial points of the boundary loops on the zero section.) Figure 7 shows the moduli space of such annuli with fixed finite conformal modulus. It is an interval over which the interior marked point moves from one boundary component to the other. Each end of the interval corresponds to a rigid nodal curve consisting of an annulus with one boundary marked point and a disc with an interior and a boundary marked point, where the marked point and the node are aligned in the annulus, and the two marked points and the node are aligned in the disc (i.e. they lie on a circle segment perpendicular to the boundary). Figure 8 shows the moduli space of such annuli with varying conformal modulus. It is a pentagon in which we will view the two lower sides as being "horizontal" (although they meet at an actual corner). Then, in the vertical direction the conformal modulus increases from 0 (on the top side) to ∞ (on the two lower sides), while in the horizontal direction the interior marked point moves from one boundary component to the other. In all configurations the marked points and nodes are aligned. The interior nodes occurring along  the bottom sides carry asymptotic markers (depicted as arrows) that are aligned with the boundary marked points. In particular, each interior node comes with an orientation reversing isomorphism between the tangent circles matching the asymptotic markers (this is the "decorated compactification").

Floer annuli
Now, we define a moduli space of Floer maps into T * M over the moduli space P of annuli in Fig. 8. For this, we choose a family of 1-forms β τ , τ ∈ P, with the following properties (see Fig. 9): • dβ τ 0 for all τ ; The annuli carry two marked points on their boundary circles (depicted as black dashes) which are aligned with the interior puncture. Again, all interior punctures carry asymptotic markers (not drawn) that are aligned with the boundary marked points, also over broken curves and are matching across each pair of positive/negative punctures.
Note that the bottom corner of the pentagon in Fig. 8 has been replaced by a new side over which the underlying stable domain is fixed, but the weights at the positive/negative puncture vary as depicted with a ∈ [−1, 2]. Thus, the conformal modulus is 0 along the top side, and ∞ along the three bottom sides.
We fix a nonnegative Hamiltonian K : T * M → R as in Sect. 3.2. For τ ∈ P, we denote by Σ τ the corresponding (possibly broken) annulus with one positive interior puncture z + and two numbered boundary marked points z 1 , z 2 on the boundary components C 1 , C 2 , equipped with the 1-form β τ . Given x ∈ F C * (K), we define the moduli space where the condition u(z + ) = x is understood as being C ∞ -convergence u(s, ·) → x as s → ∞ in cylindrical coordinates (s, t) ∈ [0, ∞) × S 1 near the positive puncture z + . By Anosov [10], the restriction u| Ci can be uniquely parametrized over [0, 1] as an H 1 -curve proportionally to arclength such that time 0 corresponds to the marked point z i , i = 1, 2. Viewing u| Ci with these parametrisations thus yields a boundary evaluation map Note that this map is also canonically defined over the boundary of P. Indeed, this is clear everywhere except possibly over the two vertical sides where one boundary loop is split into two. There one component of Σ τ is an annulus without interior puncture, on which the map u is therefore constant (see the next subsection). Hence, in the split boundary loop, one component is constant, and we map it simply to the other component parametrized proportionally to arclength.
where χ(Σ τ ) = −1 is the Euler characteristic of the punctured annulus. However, the moduli space P(x) is not transversely cut out over the vertical sides of P. Indeed, the moduli space of non-punctured annuli appearing there has Fredholm index nχ(A)+1 = 1, where χ(A) = 0 is the Euler characteristic of the annulus A and the +1 corresponds to the varying conformal modulus. But the actual dimension of this space is n + 1, where n is the dimension of the space of constant maps A → M . In the following subsections, we explain how to achieve transversality by perturbing the Floer equation by a section in the obstruction bundle.

Moduli problems and obstruction bundles
To facilitate the discussion in the next subsection, we introduce in this subsection a general setup for moduli problems and obstruction bundles. Our notion of a moduli problem will be a slight generalisation of that of a Gmoduli problem in [18] for the case of the trivial group G, which allows us to work with integer rather than rational coefficients. A moduli problem is a quadruple (B, F, S, Z) with the following properties: • p : F → B is a Banach fibre bundle over a Banach manifold; • Z ⊂ F is a Banach submanifold transverse to the fibres 3 ;

M:=S −1 (Z) ⊂ B
is compact and for each b ∈ M the composed operator

Moreover, the linear operators
such that the resulting isomorphism from det(S) to det(S ) is orientation preserving. Proof. This follows directly from the corresponding results in [18]. To construct the Euler class, we compactly perturb S to a section S which is transverse to Z; then M = S −1 (Z) is a compact manifold of dimension d = ind(S) which inherits a canonical orientation and thus represents a class in H d (B; Z), and it is easy to see that this class is independent of the choice of perturbation. The assertion about morphisms is obvious.
A special case of a moduli problem arises if F = E → B is a Banach vector bundle and Z = Z E is the zero section in E. In this case D b S is the vertical differential of S at b ∈ M = S −1 (0) and we arrive at the usual notion of a Fredholm section. This is the setup considered in [18]; the general case can be reduced to this one (via a morphism of moduli problems) by passing to the normal bundle of Z.
Consider Proof. Choose N → Z a smooth Banach vector bundle such that for each z ∈ Z, Since N represents the normal bundle to Z in F, we can assume that DS takes values in N and O is a subbundle of N complementary to im DS. Pick a fibrewise Riemannian metric on F whose exponential map restricts to a fibre preserving embedding Combined with (24) this yields a canonical isomorphism so an orientation of det(S) induces an orientation of A. In the case ind(S) = 0 this can be made more explicit as follows. Then, rk O = dim M and an orientation of det(S) induces an isomorphism preserves orientations, and −1 otherwise. Then, the signed count is the Euler number of the obstruction bundle O → M.
Finally, consider a moduli problem (B, F, S, Z) which splits as follows: Proof. Since S 1 is transverse to Z 1 , it follows that B ⊂ B is a submanifold and (B, F , S, Z) defines a moduli problem. Now, it follows directly from the definitions that (ψ, Ψ) as in the lemma induces for b ∈ B the canonical identities

Lemma 6.7. In the situation above, there exists a reduced moduli problem
hence it defines a morphism of moduli problems.

Constant Floer annuli
In this subsection, we apply the results of the previous subsection to moduli spaces of annuli. We begin with a rather general setup. Let (Σ, j) be a compact Riemann surface with boundary, and (V, J) be an almost complex manifold with a half-dimensional totally real submanifold L ⊂ V . For m ∈ N and p ∈ R with mp > 2, we consider the Banach manifold We denote its zero set by M:=∂ −1 K (0). For u ∈ M the usual energy estimate (see e.g. [34]) gives where the Hamiltonian action of u| ∂Σ vanishes because both the Liouville form and the Hamiltonian K vanish on the zero section M . This implies that du − X K (u) ⊗ β ≡ 0. Since X K vanishes near the zero section, it follows that du ≡ 0 near ∂Σ and therefore, by unique continuation, u is constant equal to a point in M . Hence, the moduli space consists of points in M , viewed as constant maps Σ → M . Since X K vanishes near the zero section, the Floer operator ∂ K agrees with the Cauchy-Riemann operator ∂ near M, so we can and will replace ∂ K by ∂ in the following discussion of obstruction bundles. We identify Σ with the standard annulus [0, R] × R/Z and its trivial tangent bundle T Σ = Σ × C. Consider a point u ∈ M , viewed as a constant map u : Σ → M . We identify Then, we have where for the last equality, we use the canonical isomorphism With these identifications, the linearized Cauchy-Riemann operator reads An easy computation using Fourier series (see [14]) shows that Constant annuli of modulus zero. Annuli of conformal modulus zero can be viewed as moduli problems in two equivalent ways. For the first view, we take as domain the compact region A ⊂ C bounded by two circles touching at one point, the node. Given (V, J) and L ⊂ V as above, we, therefore, obtain a moduli problem ( the Cauchy-Riemann operator S A = ∂ A , and the zero section For the second view, we take as domain the closed unit disc D ⊂ C with ±i viewed as nodal points which are identified. This gives rise to a moduli Note that the indices of the two moduli problems agree, Let φ : D → A be a continuous map which maps ±i onto the nodal point and is otherwise one-to-one, and which is biholomorphic in the interior. 4 Then, composition with φ defines a diffeomorphism (where we use as area form on A the pullback under φ of an area form on D). Since ev : B D → L × L is transverse to the diagonal Δ, we are in the situation of Lemma 6.7. We conclude that there exists a morphism of moduli problems where ψ : Moreover, in view of the preceding discussion and the fact that the Cauchy-Riemann operator ∂ D : B D → E D over the disc is transverse to the zero section, they both satisfy the hypotheses (i) and (ii) of Lemma 6.5, so combined with the preceding discussion, we obtain Corollary 6.9. There exists a commuting diagram of morphisms of moduli problems

Proof of Theorem 6.1
Now, we can conclude the proof of Theorem 6.1. For x ∈ F C * (K) consider the moduli space P(x) of Floer annuli described in Sect. 6.2 with its boundary evaluation map ev ∂ : P(x) → Λ × Λ. Pick a 1-form η on M with nondegenerate zeroes p 1 , . . . , p k . As in Sect. 6.4, we view η as a section of the obstruction bundle over the vertical sides of the hexagon in Fig. 9. We extend this section by a cutoff function to a section η over the whole hexagon and add it as a right hand side to the Floer equation. We choose the data such that the moduli space P(x) is transversely cut out, and thus defines a compact manifold with corners of dimension CZ(x)+2−n.
We may assume without loss of generality that M is connected. We pick a C 2 -small Morse function V : M → R with a unique maximum at q 0 ∈ M such that p 1 , . . . , p k flow to q 0 under the positive gradient flow of V . Let MC * (S) denote the Morse complex of the perturbed energy functional (note that there is no factor 1/2 in front of |q| 2 ). For x ∈ F C * (K) and a, b ∈ MC * (S), we define where W + (a) is the stable manifold of a with respect to the negative pseudogradient flow of S. Recall that the boundary evaluation map involves reparametrisation of the boundary loops proportionally to arclength. For generic choices, these are manifolds of dimension If the dimension is 0 these spaces are compact and their signed counts define a degree 2 − n map Next, we consider a 1-dimensional moduli space P dim=1 (x; a, b) and compute its boundary. Besides splitting off index 1 Floer cylinders and negative pseudo-gradient flow lines, which give rise to the term [∂, Θ 1 ], there are contributions from the sides of the hexagon in Fig. 9 which we analyse separately. Note that the indices now satisfy Vertical left side: Here, the broken curves consist of a half-cylinder attached at a boundary node to an annulus without interior puncture, where the two boundary loops flow into a, b under the negative pseudo-gradient flow of S. By the discussion in Sect. 6.4 the moduli space of annuli is [0, ∞] × η −1 (0), where [0, ∞] encodes the conformal modulus and η −1 (0) consists of the points p 1 , . . . , p k (with signs σ(p i )). In particular, we must have b = q 0 and therefore ind(b) = ind(q 0 ) = 0. The half-cylinders belong to the moduli space They carry a boundary nodal point which is aligned with the boundary marked point (0, 0) and the puncture at ∞, and is therefore given by (0, 1/2). The evaluation at the nodal point defines an evaluation map Lower left side: Here, the broken curves consist of a disc with two interior punctures, one positive and one negative, attached at its negative puncture to the positive puncture of a half-cylinder along an orbit in F C * (−K), where the two boundary loops flow into a, b under the negative pseudogradient flow of S. By choosing the 1-form β equal to dt on a long cylindrical piece of the half-cylinder, we can achieve that these half-cylinders are in one-to-one correspondence with broken curves consisting of a cylinder with weights (−1, 1) and a half-cylinder with weights (1, 2), as shown in the middle of Fig. 10. Reinterpreting these curves as on the right of that figure, we see that their count corresponds to the composition −(Γ 1 ⊗ Ψ)(1 ⊗ c F 0 ), where we recall that Γ 1 denotes the first term in the expression Γ = Γ 1 + Γ 2 + Γ 3 from (22).
Lower right side: Similarly, the contribution from the lower right side corresponds to the composition (Ψ ⊗ Γ 1 )(c F 0 ⊗ 1). The discussion so far shows that (25) where Θ top and Θ bottom are the degree 1 − n maps arising from the contributions of the top and bottom sides of the hexagon to the boundary of P dim=1 (x; a, b) which we discuss next.
Bottom side: The family of broken curves on the bottom side can be deformed in an obvious way to the family of broken curves shown in Fig. 11. Since the half-cylinders with weights (2, 2) define the map Ψ and the family of 3-punctured spheres above them defines the continuation coproduct λ F from Sect. 3.2, this shows that Θ bottom is equal to −(Ψ ⊗ Ψ)λ F .
Top side: The family on the top side of the hexagon consists of punctured annuli of modulus 0, i.e. punctured discs with two nodal points on the boundary that are identified to a node. Moreover, the boundary carries two marked points that are separated by the nodal points and aligned with the interior puncture. We wish to relate this family to the loop coproduct, but Figure 11. Degenerating the curves on the bottom side for this we face two problems: First, the boundary loops carry two marked points whereas the loops for the loop coproduct carry only one (the initial time t = 0); and second, the self-intersection of the boundary loop occurs at the nodal points and not at one of the marked points.
Both problems are resolved simultaneously as follows. We enlarge this 1-parametric family to a 2-parametric family in which we keep the two boundary marked points aligned, but drop the condition that the interior puncture is aligned with them. The 2-parametric family forms the hexagon shown in Fig. 12. Here, the interior puncture is depicted as a cross, the aligned boundary marked points as endpoints of a dashed line, and the nodal points as thick dots. The bottom side of the hexagon (drawn in black) corresponds to the 1-parametric family on the top side of Fig. 9. Note that here we made a choice by letting the interior puncture move freely above the dashed line connecting the two boundary marked points; we could equally well have taken the mirror hexagon where the interior puncture moves below the dashed line.
The hexagon in Fig. 12 defines a deformation from the bottom (black) side to the top side (drawn in red). The configurations in this figure are to be interpreted as follows.
• Each configuration has two boundary loops obtained by going around in the counterclockwise direction: the first loop from the bottom to the top nodal point, and the second one from the top to the bottom nodal point. Each boundary loop carries a marked point. As before, each boundary loop of the zero section is reparametrized proportionally to arclength and then flown into a critical point on Λ under the negative pseudo-gradient flow of the functional S : Λ → R.
• In each configuration, the unique component carrying the interior puncture (which may be nonconstant) is drawn as a large disc, so the small discs are all constant. In particular, each small disc carrying the two nodal points is a constant annulus of modulus zero. Under the perturbation of the Cauchy-Riemann equation described in Corollary 6.
Thus, the hexagon in Fig. 12 provides a chain homotopy Θ 2 from Θ top (defined by the bottom side) to the operation • Consider now the top side. Since both marked points and the black nodal point lie on the same constant component, we can remove this component and replace the three points by one nodal/marked point as shown in Fig. 13.
The boundary of these configurations consists of loops q : [0, 1] → M with one (black) marked/nodal point at time 0 and an additional (red) nodal point at time s which moves from 0 to 1 as we traverse the side from left to right. In view of Corollary 6.9 and Remark 6.6, the map Θ top : F C * (K) → MC * (S)⊗MC * (S) is defined by counting isolated configurations consisting of punctured discs as in the definition of the moduli spaces M(x) from Sect. 5.5, additionally decorated with two marked points, with an incidence condition at the marked points, followed by semi-infinite negative pseudo-gradient lines of S starting at the de-concatenated loops. Now, we deform Θ top once more by inserting a negative pseudo-gradient trajectory of S of finite length T ≥ 0 between the boundary loop of the disc and the loop on which we impose the incidence condition at the marked points. As T → ∞ this becomes the chain map Ψ : F C * (K) → MC * (S) followed by the Morse theoretic coproduct λ, whereas on the boundary of the top side we see appear the terms (Ψ ⊗ , with Γ 3 being the third term from (22). We obtain therefore a homotopy Θ 3 between the operation Θ top defined by the top side and λΨ [∂, Summing together Eqs. (25), (26) and (27), and recalling that Γ = Γ 1 + Γ 2 + Γ 3 , we obtain the desired relation (23).
For n = 2, the condition Θc F = 0 follows by an index argument analogous to the proof of the relation λ F c F = 0 in Proposition 3.1. Together with the discussion at the beginning of this section, this concludes the proof of Theorem 6.1.
Remark 6.10. (Perturbation by 1-form/vector field) Let us analyse how the perturbation by a 1-form η with transverse zeroes propagates to the diagrams in the preceding proof. Along the left hand sides of the hexagon in Fig. 9, we perturb the Floer operator by η at the second output. This continues along the left hand sides of the hexagon in Fig. 12 as the perturbation of the constant modulus zero annuli by the vector field v corresponding to η at the second output. As a result, the left hand configuration in Fig. 13 is perturbed by applying the time-one-map f of v as we go counterclockwise from the black to the red dot. This means for s 0 close to 0 the evaluations of the corresponding loops q : [0, 1] → M at time 0 (the black dot) and s (the red dot) are related by q(s) = f (q(0)). Along the right hand sides of the hexagon in Fig. 9, we perturb the Floer operator by η at the first output. As a result, the right hand configuration in Fig. 13 is perturbed by applying the time-one-map f of v as we go clockwise from the black to the red dot. This means that for s 1 close to 1 the evaluations of the corresponding loops q : [0, 1] → M at time 0 (the black dot) and s (the red dot) are related by q(0) = f (q(s)), or equivalently, q(s) = f −1 (q(0)). Therefore, the perturbation by the 1-form η on the Floer side translates on the loop side into the perturbation by an s-dependent vector field which agrees with v near s = 0 and with −v near s = 1.

Relation to other Floer-type coproducts
The continuation coproduct λ F discussed in the previous sections descends to positive action symplectic homology SH >0 * (D * M ) (since the action inequality implies that if the input orbit is constant, then so must be the output orbits).
In this section we relate λ F to other coproducts on SH >0 * (D * M ) that have appeared in the literature, thus proving Theorem 1.1 from the Sect. 1.
In particular, we will prove that λ F agrees with the Abbondandolo-Schwarz coproduct λ AS defined in [4]. Abbondandolo and Schwarz defined in [4]  −→ H * (Λ, Λ 0 ) intertwines the coproduct λ AS with the loop coproduct λ. However, to our knowledge no proof of this result has appeared. We will actually give two proofs in this section: the first one uses Theorem 6.1 and the identification λ F = λ AS , the second one uses a direct argument and suitable interpolating moduli spaces.
This section is structured as follows. In Sect. 7.1, we recall from [20] the definition of the varying weights coproduct λ w , which coincides with λ F by [20,Lemma 7.2] and which can be more easily related to λ AS . In Sect. 7.2, we recall from [4] the definition of the Abbondandolo-Schwarz coproduct λ AS . In Sect. 7.3, we show that λ AS is equal to λ w . In Sect. 7.4, we prove directly that λ AS corresponds to the loop coproduct λ under the isomorphism SH >0 The situation is summarised in the following diagram.
λ §7.4 The whole discussion concerns the free loop space, but it carries over verbatim to the based loop space.
For simplicity, we assume throughout this section that M is oriented and we use untwisted coefficients in a commutative ring R; the necessary adjustments in the nonorientable case and with twisted coefficients are explained in Appendix A. We denote S 1 :=R/Z and Λ:=W 1,2 (S 1 , M).

Varying weights coproduct
We recall the definition of the varying weights coproduct λ w on SH >0 * (V ) from [20, §7.1]. Since there we actually describe the algebraically dual product on SH <0 * (V, ∂V ), we will recap in some detail the necessary notation and arguments. The construction goes back to Seidel, see also [26]. We work with a Liouville domain V of dimension 2n, the symplectic completion is denoted V = V ∪[1, ∞)×∂V and the radial coordinate in the positive symplectisation [1, ∞) × ∂V is denoted r.
Let Σ be the genus zero Riemann surface with three punctures, one of them labelled as positive χ + and the other two labelled as negative υ − , ζ − , endowed with cylindrical ends [0, ∞) × S 1 at the positive puncture and (−∞, 0] × S 1 at the negative punctures. Denote (s, t), t ∈ S 1 the induced cylindrical coordinates at each of the punctures. Consider a smooth family of 1-forms β τ ∈ Ω 1 (Σ), τ ∈ (0, 1) satisfying the following conditions: • (nonpositive) dβ τ ≤ 0; In the symplectisation [1, ∞) × ∂V , we have H ≥ 0 and therefore d(Hβ) ≤ 0, so that elements of the above moduli space are contained in a compact set. The dimension of the moduli space is When it has dimension zero the moduli space M 1 dim=0 (x; y, z) is compact. When it has dimension 1 the moduli space M 1 dim=1 (x; y, z) admits a natural compactification into a manifold with boundary Here, M 1 τ =1 (x; y, z) and M 1 τ =0 (x; y, z) denote the fibres of the first projection M 1 dim=1 (x; y, z) → (0, 1), (τ, u) → τ near 1, respectively near 0. (By a standard glueing argument the projection is a trivial fibration with finite fibre near the endpoints of the interval (0, 1).) Consider the degree −n + 1 operation defined on generators by where #M 1 dim=0 (x; y, z) denotes the signed count of elements in the 0-dimensional moduli space M 1 dim=0 (x; y, z). Consider also the degree −n operations where #M 1 τ =i (x; y, z) denotes the signed count of elements in the 0-dimensional moduli space M 1 τ =i (x; y, z). Denote by ∂ F the Floer differential on the Floer complex of H. The formula for ∂M 1 dim=1 (x; y, z) translates into the algebraic relation We now claim that To prove the claim for λ w 0 , note that this map can be expressed as a composition (c⊗id)•λ 0 , where λ 0 : F C * (H) → F C * (τ H)⊗F C * (H) is a pair-of-pants coproduct with τ > 0 small, and c : F C * (τ H) → F C * (H) is a continuation map. Taking into account that τ H has no nontrivial 1-periodic orbits for τ small, and because the action decreases along continuation maps, we obtain c(F C * (τ H)) ⊂ F C =0 * (H), which proves the claim. The argument for λ w 1 is similar.
It follows that λ w induces a degree −n + 1 chain map Passing to the limit as the slope of H goes to +∞, we obtain the degree −n + 1 varying weights coproduct λ w on SH >0 * (V ).

Abbondandolo-Schwarz coproduct
In this subsection, we recall from [4] the definition of a secondary pair-ofpants product on Floer homology of a cotangent bundle, which we will refer to as the Abbondandolo-Schwarz coproduct λ AS . We recall the notation and conventions from Sect. 5.1 regarding the Floer complex. In particular near the zero section H(q, p) = ε|p| 2 + V (q) for a small ε > 0 and a Morse function V : M → R such that all nonconstant critical points of A H have action larger than min V . For x, y, z ∈ Crit(A H ), set (see Fig. 14) Note that the matching conditions imply u(0, τ) = u(0, 0).
Here, C : T * M → T * M is the antisymplectic involution (q, p) → (q, −p), Δ ⊂ M × M is the diagonal, and N * Δ ⊂ T * (M × M ) its conormal bundle. The space M 1,AS (x; y, z) is a moduli space with jumping Lagrangian boundary conditions as in [3], so for generic H and J it is a transversely cut out manifold. Its dimension is given by the Fredholm index of the linearised problem [4, (37)].
If M 1,AS (x; y, z) has dimension zero it is compact and defines a map If it has dimension 1, it can be compactified to a compact 1-dimensional manifold with boundary Here, the first three terms correspond to broken Floer cylinders and the last two terms to the intersection of M 1,AS (x; y, z) with the sets {τ = 1} and {τ = 0}, respectively. Therefore, we have where for i = 0, 1, we set Let us look more closely at the map λ AS 1 . For τ = 1 the matching conditions in M 1,AS (x; y, z) imply that w(0, t) = u(0, 0) is a constant loop. For action reasons z must then be a critical point, so that Im(λ AS , and therefore λ AS descends to a chain map Note first that for 0-dimensional moduli spaces M 1,AS dim=0 (x; y, z), we can restrict τ to (0, 1). Given τ ∈ (0, 1) a triple (u, v, w) as in the definition of M 1,AS (x; y, z) can be interpreted as a single mapũ : Σ → T * M satisfying (dũ − X H ⊗ β τ ) 0,1 = 0, where Σ is a Riemann surface and β τ is a 1-form explicitly described as follows. The Riemann surface is closed 1-form dt. 6 Upon identifying the cylindrical ends at the negative punctures with (−∞, 0] × S 1 , this canonical 1-form becomes equal to τ dt, respectively (1 − τ )dt at those punctures. The 1-form β τ is defined to be the discontinuous 1-form equal to dt on the cylindrical end [0, ∞) × S 1 at the positive puncture, equal to 1 τ dt on the cylindrical end (−∞, 0]×R/τ Z at the first negative puncture, and equal to 1 1−τ dt on the cylindrical end (−∞, 0]×R/(1−τ )Z at the second negative puncture. Equivalently, upon normalising the cylindrical ends at the negative punctures into (−∞, 0] × S 1 , the 1-form β τ is simply dt. This discontinuous 1-form β τ can be interpreted as a limit of 1-forms which are obtained by interpolating from τ dt and (1 − τ )dt (near 0) towards dt (near −∞) in the normalised cylindrical ends at the negative punctures, where the interpolation region shrinks and approaches s = 0. It was noted in Sect. 7.2 that the limit case defines a Fredholm problem M 1,AS (x; y, z) with jumping Lagrangian boundary conditions. The Fredholm problem before the limit is naturally phrased in terms of the Riemann surface Σ without boundary, but it can be reinterpreted as a problem with Lagrangian boundary conditions by cutting Σ open along {s = 0}. As such, it converges in the limit to the Fredholm problem with jumping Lagrangian boundary conditions described above. By regularity and compactness, the two Fredholm problems are equivalent near the limit, and the corresponding counts of elements in 0-dimensional moduli spaces are the same.

Abbondandolo-Schwarz coproduct equals loop coproduct
Recall the Hamiltonian H : S 1 ×T * M → R from Sect. 7.2 and its fibrewise Legendre transform L : S 1 × T M → R from Sect. 2.2. Also recall from Sect. 2.2 the notations concerning the Morse complex MC * of the action functional S L which we will use freely. In particular, ∂ denotes the Morse boundary operator and λ the coproduct from Remark 2.12.
We assume that M is oriented and we use the Morse complex twisted by the local system σ obtained by transgressing the second Stiefel-Whitney class.
Following [4], for x ∈ Crit(A H ) and a ∈ Crit(S L ), we define where W + (a) is the stable manifold of a for the negative pseudo-gradient flow of S L . See Fig. 4.  The induced map on homology is an isomorphism intertwining the pair-of-pants product with the loop product.
Proposition 7.4. The map Ψ descends to an isomorphism on homology modulo the constant loops which intertwines the Abbondandolo-Schwarz coproduct λ AS with the loop coproduct λ.
where M(x) was defined above and φ s : Λ → Λ for s 0 denotes the flow of the negative pseudo-gradient of S L . Note that α, β, γ in the definition of M − (x; b, c) are actually redundant and just included to make the definition more transparent. As above it follows that for generic H these spaces are transversely cut out manifolds of dimensions dim M + (x) = CZ(x) − n + 2 and We set If this space has dimension zero it is compact and defines a map If it has dimension 1 it can be compactified to a compact 1-dimensional manifold with boundary where M 1 (a; b, c) are the moduli spaces in Remark 2.12 defining the coproduct λ with f t = id. Here, the first term corresponds to splitting off of Floer cylinders, the second and third ones to splitting off of Morse pseudo-gradient lines, the fourth one to σ = +∞, the fifth one to σ = −∞, and the last two terms to the intersection of M 2 (x; b, c) with the sets {τ = 1} and {τ = 0}, respectively. The intersections of M ± (x; b, c) with the set {σ = 0} are equal with opposite orientations and thus cancel out. Therefore, we have where for i = 0, 1, we set Arguing as in the previous subsection, we see that the Θ 0 has image in MC =0 * ⊗ MC * , and Θ 1 has image in MC * ⊗ MC =0 * . Together with Eq. (34) this shows that Θ descends to a map Θ : F C >0 * → (MC >0 ⊗ MC >0 ) * −n+2 between the positive chain complexes which is a chain homotopy between (Ψ ⊗ Ψ)λ AS and λΨ, which concludes the proof.

Loop coproduct for odd-dimensional spheres
In this section, we compute the loop coproduct on reduced loop homology H * (ΛS n ) = H * (ΛS n ) of odd-dimensional spheres S n . For its definition, we use a Morse function S n → R with only two critical points, the minimum and the maximum, and a vector field v (or equivalently a 1-form η) which is nowhere vanishing. By Proposition 4.7, the coproduct does not depend on these choices if n 2. For the same reason, in the definition of the loop coproduct, we can use a constant family v τ ≡ v instead of the family v τ interpolating between v 0 = v and v 1 = −v from Sect. 2.2 (see Remark 4.8).
In the case n = 1, we will see that the loop coproduct actually depends on the choice of v.
For our computation, we first give a third definition of the loop coproduct on reduced loop homology in terms of singular homology.

Topological description of the loop coproduct
We define the loop coproduct on singular loop homology relative to χ·point. It is induced by a densely defined operation on singular chains constructed as follows. The beginning of the construction is like in Sect. 2.2. We fix a small vector field v on M with nondegenerate zeroes such that the only periodic orbits of v with period 1 are its zeroes, and we consider a generic family of vector fields v τ , τ ∈ [0, 1] which interpolates For each q ∈ M , we denote as in Sect. 2.2 the induced path from q to f τ (q) by π τ q : [0, 1] → M , π τ q (t):=f τ t (q), and the inverse path by (π τ q ) −1 .
In the spirit of [13], let a : K a → Λ be a chain such that the map } is a compact manifold with corners and we define Fig. 1 where α = a(x). At τ = 0 and τ = 1 the condition in K λ(a) becomes a(x)(0) = q ∈ Fix(f 0 ), respectively a(x)(1) = q ∈ Fix(f 1 ), and denoting the constant loop at q by the same letter we find It follows that where q is viewed as a 0-chain and the loop products with the constant loop q are given by Here, the signs ind ±v (q) arise from the discussion before Remark 2.8, noting that the restriction of ev a to τ = 0 or τ = 1 is the composition of the evaluation K a → M , x → a(x)(0) and the map M → M ×M , q → f 0 (q), q , respectively q → f 1 (q), q .
Let us now fix a basepoint q 0 ∈ M and consider a such that the map ev a,0 : is transverse to q 0 . We choose all zeroes of v (i.e. fixed points of f 0 and f 1 ) so close to q 0 that ev a,0 is transverse to each of them. Then, after identifying the domains K q•a with K q0•a and transferring loops at q to loops at q 0 , we have where is the Euler characteristic of M . Recalling the notation C * (Λ, χpt):=C * (Λ)/χRq 0 for the chains relative to χ·point, we see that λ induces a chain map C * Λ → (C(Λ, χpt) ⊗ C(Λ, χpt)) * +1−n . Moreover, this factors through C * (Λ, χpt): if n ≥ 2 this holds for degree reasons, and if n = 1 this holds tautologically because the Euler characteristic is zero. The outcome is a coproduct H * (Λ, χpt) → (H(Λ, χpt)⊗H(Λ, χpt)) * +1−n on the homology relative to χ·point. Under our standing assumption of orientability on M , this is the same as a coproduct on reduced loop homology H * Λ → (HΛ ⊗ HΛ) * +1−n .

Loop coproduct for spheres of odd dimension n 3
In this subsection, we use Z-coefficients and assume n 3 is odd. Recall from [23] that the degree shifted homology of the free loop space of S n is the free graded commutative algebra where the shifted degree |a| is related to the geometric degree by |a| = deg a− n. Here, A is the class of a point (of geometric degree 0) and U is represented by the descending manifold of the Bott family of simple great circles tangent at their basepoint to a given non-vanishing vector field on the sphere (of geometric degree 2n − 1). Since χ(S n ) = 0, the coproduct λ is defined on H * (ΛS n ) and has shifted degree 1−2n (and geometric degree 1−n). The unit 1 is represented by the fundamental chain of all constant loops (of geometric degree n).
We begin with some explicit computations of coproducts, to be compared to [29].
Proof. We will actually prove these relations in H * (ΛS n ), in which case (a-c) remain unchanged, but (d) becomes λ(U ) = A ⊗ 1 + 1 ⊗ A (the sign change comes from the odd degree shift by n). We recall the observation made at the beginning of this section that in the definition of the loop coproduct, we can use a constant family of vector fields v τ ≡ v. We fix such a choice in the sequel, with v small and nowhere vanishing. We denote f τ t = f t the flow of v τ = v, and we denote f τ = f the time-one flow.
(a) To prove λ(A) = 0, we represent A by the constant loop at q 0 . Then, f τ (q 0 ) = q 0 for all values of τ ∈ [0, 1] and therefore λ(A) is supported by the empty set. To prove λ(1) = 0, we represent 1 by the S n -family of constant loops and note that f τ (q) = q for all q ∈ S n . Thus, λ(1) is supported by the empty set.
(b) We fix a unit tangent vector v 0 at q 0 and we represent AU by the (n − 1)-chain a : K n−1 → ΛS n of all circles with fixed initial point q 0 and initial direction v 0 . (K n−1 is the (n − 1)-disc of all 2-planes in R n+1 through q 0 containing the vector v 0 , whose boundary is mapped to q 0 .) Then, a(k)(0) = q 0 for all k ∈ K n−1 . Since the evaluation map (k, τ ) → a(k)(τ ) covers S n once, there exists a unique (k, τ ) for which a(k)(τ ) = f τ (q 0 ) = f (q 0 ). Therefore, λ(a) is homologous to the 0-cycle A ⊗ A.
(c) We represent AU 2 by the (2n − 2)-chain a : K 2n−2 → ΛS n of all circles with fixed initial point q 0 . (K 2n−2 is a fibre bundle K n−1 → K 2n−2 → S n−1 , where S n−1 is the (n−1)-sphere of all initial directions at q 0 and K n−1 is the (n−1)-disc from (b) of all circles through q 0 in a given initial direction.) Then, a(k)(0) = q 0 for all k ∈ K 2n−2 . Recall that f (q 0 ) = q 0 is a point close to q 0 . Let us fix some initial direction v 0 at q 0 . For every sufficiently large circle (whose diameter is bigger than the distance from q 0 to f (q 0 )) with initial point q 0 and initial direction v 0 there exist precisely two rotations of the initial direction such that the rotated circles pass through f (q 0 ). One of these rotated circles passes though f (q 0 ) near τ = 0 and the other one near τ = 1. As the circle varies over the (n − 1)-chain K n−1 of all circles with initial point q 0 and initial direction v 0 (and we let f (q 0 ) move to q 0 ), these two families of rotated circles give rise to cycles representing the classes A ⊗ AU and AU ⊗ A, respectively.
(d) We represent U by the (2n − 1)-chain a : K 2n−1 → ΛS n of all circles starting at their basepoint q ∈ S n in direction v(q). (K 2n−1 is a fibre bundle K n−1 → K 2n−1 → S n , where S n corresponds to the initial points and K n−1 is the (n − 1)-disc from (b).) For every q, there exists a unique circle a(x q ) starting at q in direction v(q) and passing through f (q). Since all the circles constituting the chain a are simple, there is a unique τ q such that a(x q )(τ q ) = f (q) = f τq (a(x q )(0)). By splitting each a(x q ) at the parameter value τ q using the path π q (t) = f t (q), we obtain a cycle s : S n → Λ×Λ that represents λ(a). This cycle has degree n, it sits over the diagonal Δ ⊂ S n × S n as an element of the fibration (ev, ev) : Λ × Λ → S n × S n , and denoting π : Λ × S n Λ → S n the restriction of this fibration to the diagonal, we have π • s = Id S n . On the other hand, H n (Λ × S n Λ) has rank 1, generated by the class of the diagonal: that the rank is at most 1 follows by inspection of the spectral sequence of the fibration ΩS n × ΩS n → Λ × S n Λ → S n , using the fact that H * (ΩS n ) is a polynomial ring on one generator in degree n − 1, and that it is at least one follows from the fact that the diagonal is a section. This implies that the cycle s is homologous to the diagonal Δ ⊂ S n × S n in Λ × S n Λ, hence also in Λ × Λ, and we conclude λ Remark 8.4. We note in particular that this extended coproduct on reduced homology has contributions from the constant loops, unlike the one from [29]. These contributions from the constant loops play an essential role for the unital infinitesimal algebra structure.
The previous computation allows us to recover the Sullivan-Goresky-Hingston coproduct on H * (ΛS n , Λ 0 ) = H * +n (ΛS n , Λ 0 ) [27,29]. Our method ultimately relies on the infinitesimal relation and involves a minimal geometric input in the form of Lemma 8.1 (a) and (d). In comparison, the computation from [29] of the coproduct on H * (ΛS n , Λ 0 ) relies on geometric input which is quite involved. In a sense, the "algebra" of the infinitesimal relation replaces the "geometry" of spaces of circles from [29]. Corollary 8.5. For n ≥ 3 odd, the Sullivan-Goresky-Hingston coproduct on H * (ΛS n , Λ 0 ) is given by Proof. It is enough to discard the terms involving constant loops from the formulas of Proposition 8.3.

Loop coproduct for S 1
In this section, we study the loop coproduct on the loop space of S 1 = R/Z. The degree shifted loop homology with R-coefficients is as a ring with respect to the loop product given by where the classes AU k and U k are represented by the cycles Proposition 8.6 shows that for M = S 1 the coproduct on reduced loop homology does depend on the choice of a nowhere vanishing vector field. One can verify that both coproducts λ ± define together with the loop product a commutative cocommutative infinitesimal anti-symmetric bialgebra in the sense of [21] with The unital infinitesimal relation reads now Remark 8.7. Just like in the case of higher dimensional spheres, the expressions of the coproducts λ ± on H * (ΛS 1 ) can be derived from the unital infinitesimal relation combined with knowledge of the product μ and of the values For example, to compute λ ± (U −1 ) one applies the unital infinitesimal relation to U ⊗U −1 , to compute λ ± (AU −1 ) one applies the unital infinitesimal relation to A ⊗ U −1 (or to AU −1 ⊗ U ) etc.
Remark 8.8. The example of the circle is very rich in that it also shows that the condition v 1 = −v 0 for the family of vector fields v τ is necessary in order for the coproducts to have a good algebraic behaviour. For example, with a constant family v τ ≡ v + , we find an operation λ v+,v+ given by k<0.
A direct check shows that this operation is neither coassociative, nor cocommutative, though it satisfies the unital infinitesimal relation with λ v+,v+ (1) = 0, i.e. Sullivan's relation. Similarly, with the constant family v τ ≡ v − , we find an operation λ v−,v− given by Again, this is neither coassociative, nor cocommutative, though it satisfies Sullivan's relation.

A.1. Conventions
We use the following conventions from [7, §9.7]. Given a finite-dimensional real vector space V , its determinant line is the 1-dimensional real vector space det V = Λ max V . We view it as being a Z-graded real vector space supported in degree dim R V . To any 1-dimensional graded real vector space L, we associate an orientation line |L|, which is the rank 1 graded free abelian group generated by the two possible orientations of L, modulo the relation that their sum vanishes. The orientation line |L| is by definition supported in the same degree as L. When L = det V , we denote its orientation line |V |. Given a Z-graded line (rank 1 free abelian group), its dual line −1 = Hom Z ( , Z) is by definition supported in opposite degree as . There is a canonical isomorphism −1 ⊗ ∼ = Z induced by evaluation.
Given a Z-graded object F , we denote F [k] the Z-graded object obtained by shifting the degree down by k ∈ Z, i.e. F [k] n = F n+k . For example, the shifted orientation line |V |[dim V ] is supported in degree 0. A linear map f : E → F between Z-graded vector spaces or free abelian groups has degree d if f (E n ) ⊂ F n+d for all n. In an equivalent formulation, the induced map f [d] : E → F [d] has degree 0. For example, the dual of a vector space or free abelian group supported in degree k is supported in degree −k. This is compatible with the grading convention for duals of Z-graded orientation lines. Given a Z-graded rank 1 free abelian group , we denote the same abelian group with degree set to 0. For example Given two oriented real vector spaces U and W , we induce an orientation on their direct sum U ⊕ W by defining a positive basis to consist of a positive basis for U followed by a positive basis for W . This defines a canonical isomorphism at the level of orientation lines Given an exact sequence of vector spaces we induce an orientation on V out of orientations of U and W by defining a positive basis to consist of a positive basis for U followed by the lift of a positive basis for W . This defines a canonical isomorphism The following example will play a key role in the sequel. This gives rise to a canonical isomorphism |Δ| ⊗ |νΔ| ∼ = p * |M | ⊗ p * |M | and, because p * |M | ⊗ p * |M | is canonically trivial, we obtain a canonical their tensor product is given by their sum ν 1 + ν 2 . Operations like cap or (pairs of loops with the same basepoint), and . The maps f and f s are smooth and transverse to the diagonal Δ, so that F and F s are Hilbert submanifolds of codimension n. Denoting νF and νF s their normal bundles, we obtain canonical isomorphisms νF ∼ = f * νΔ, νF s ∼ = f * s νΔ. In view of Example A.1, we infer canonical isomorphisms where, in the first formula, ev 0 : F → M is the evaluation of pairs of loops at their common origin.
Proof. The first condition is c * s i * (p * 1 ν ⊗ p * 2 ν) i * s ν. Since g s is a homeomorphism, this is equivalent to g * s c * s i * (p * 1 ν ⊗ p * 2 ν) g * s i * s ν. In view of c s g s = Id Fs , this is the same as the second condition.
Definition A.4. A degree 0 local system ν on Λ is compatible with products if it satisfies the equivalent conditions of Lemma A.3.
A local system ν which is compatible with products must necessarily have degree 0 (and rank 1). Also, ν| M must be trivial: restricting both sides of (37) or (38) to the constant loops yields ν| M ⊗ ν| M ν| M .
Remark A. 5. Local systems which are compatible with products play a key role in the sequel definition of the loop product and loop coproduct with local coefficients. Condition (37) is the one that ensures the coproduct is defined with coefficients twisted by ν, whereas condition (38) is the one that ensures the product is defined with coefficients twisted by ν. That the two conditions are equivalent can be seen as yet another instance of Poincaré duality for free loops.
We refer to Remark A.11 for an additional condition on the isomorphisms (38) which is needed for the associativity of the product and coassociativity of the coproduct. We denote the corresponding local system on Λ also by τ c . Degree 0 local systems obtained in this way are called transgressive [7].
Define the local systemõ to be trivial on the components where γ preserves the orientation, and equal to ev * 0 |M | on components where γ reverses the orientation. The local systemõ is compatible with products: the equality w(γ 1 ) + w(γ 2 ) = w(g s (γ 1 , γ 2 )) holds in Z/2 for all (γ 1 , γ 2 ) ∈ F. Note that the local systemõ is not transgressive and, in case M is nonorientable, it is nontrivial on all connected components Λ α M whose elements reverse orientation. Question A.8. Characterise in cohomological terms the local systems on Λ which are compatible with products. For example, it follows from [9, Lemma 1] that, on a simply connected manifold, a local system ν is compatible with products if and only if ν| M is trivial. A mild generalisation is given by [9,Proposition 10].

A.6. Loop product with local coefficients
Following [27], we view the loop product as being defined by going from left to right in the diagram where g = i s g s for some fixed s ∈ (0, 1). More precisely, the loop product with integer coefficients is defined as the composition The first map is the homology cross-product corrected by a sign = (−1) n(i+n) ([29, Appendix B]), the second map is the composition of the map induced by inclusion Λ × Λ → (Λ × Λ, Λ × Λ\F) with excision and the tubular neighbourhood isomorphism, and the third map is the Thom isomorphism. In case M is not orientable the loop product does not land in homology with integer coefficients and thus fails to define an algebra structure on H * (Λ; Z). This can be corrected by using at the source homology with local coefficients.
The description of the maps is the same as above, with = (−1) ni because of the shift H i (Λ; μ) = H i+n (Λ; μ). However, one still needs to check that the local systems of coefficients are indeed as written. For the first, second and third map the behaviour of the coefficients follows general patterns. For the last map, we use that (p * 1 μ ⊗ p * 2 μ)| F ⊗ ev * 0 |M | g * μ, which is true for our specific μ = ev * 0 |M | −1 .

A.7. Loop coproduct with coefficients
Again following [27], we view the primary coproduct on loop homology as being defined by going from left to right in the diagram Λ ← F s cs −→ Λ × Λ for some fixed s ∈ (0, 1), where c s stands for ic s in the notation of §A.5.
We restrict in this section to coefficients in a field K and all local systems are accordingly understood in this category. The reason for this restriction is explained below. The primary coproduct with constant coefficients is defined as the composition The first map is the composition of the map induced by inclusion Λ → (Λ, Λ\ F s ) with the excision isomorphism towards the homology rel boundary of a tubular neighbourhood of F s . The second map is the Thom isomorphism. For the third map, we use that c * s p * 1 ev * 0 = ev * 0 . The fourth map is the Alexander-Whitney diagonal map followed by the Künneth isomorphism. 7 Just like for the loop product, we see that if M is nonorientable the primary coproduct fails to define a coalgebra structure on H * (Λ; K). This is corrected using homology with local coefficients as follows.
With our grading conventions, this coproduct has degree 0. Taking into account that o = ev * 0 |M | is supported in degree n, this results in the coproduct having the usual degree −n in ungraded notation. In the orientable case it recovers the usual primary coproduct.
The cohomology product with twisted coefficients is associative. It is also graded commutative when viewing it as a degree 0 product on H * −1 (Λ, Λ 0 ; ν ⊗ μ). The fundamental local system η is supported in degree −n. Our previous discussion shows that the loop product is defined and has degree 0 on H * (Λ; η), and the loop coproduct is defined and has degree +1 on H * (Λ, Λ 0 ; η −1 ). We can view the loop product as being defined on H * (Λ; η), where it has degree −n, and the loop coproduct as being defined (with field coefficients) on H * (Λ, Λ 0 ; η), where it has degree 1 − n. This point of view is useful when considering H * (Λ; η), which is a common space of definition (to which the product descends and the coproduct extends).
As proved in [5,7], the chain map Ψ = Ψ quadratic discussed in Sect. 5.3 associated to a quadratic Hamiltonian acts as Our filtered chain map Ψ = Ψ linear from Sect. 5.5 associated to a linear Hamiltonian is a chain isomorphism F C * (H) −→ MC ≤μ * (E 1/2 ; η), with μ the slope of the Hamiltonian. Given any local system ν, we obtain a filtered chain isomorphism F C * (H; ν) −→ MC ≤μ * (E 1/2 ; ν ⊗ η). In case the local system ν is compatible with products, the arguments of [1,3,7] adapt in order to show that the map Ψ intertwines the pair-ofpants product on the symplectic homology side with the homology product on the Morse side. The arguments of Theorem 6.1 adapt in order to show that the map Ψ descends on homology relative to the constant loops, where it intertwines the continuation coproduct with the loop coproduct (with field coefficients). There are also reduced versions of the map Ψ which intertwine the product, and which also intertwine the coproducts provided both are defined using the same continuation data at the endpoints. The statements for the coproducts can, moreover, be interpreted as dual statements about products in cohomology.