The string topology coproduct on complex and quaternionic projective space

On the free loop space of compact symmetric spaces Ziller introduced explicit cycles generating the homology of the free loop space. We use these explicit cycles to compute the string topology coproduct on complex and quaternionic projective space. The behavior of the Goresky-Hingston product for these spaces then follows directly.


INTRODUCTION
The central idea in Morse theory is to study the interaction between the critical sets of a function on a differentiable manifold and the topology of this manifold.While it is usually easy to understand the local homology around a critical level, it is a hard question to determine if and how all of the homology of the manifold can be understood by the individual homologies of the critical sets.In [Zil77] Ziller defines cycles on the free loop space of a compact globally symmetric space which can be used to show that the relative cycles from level homology can be completed in the free loop space.This idea goes back to Bott's K-cycles, see [Bot56] and [Bot58] as well as Bott's and Samelson's work in [BS58].Hingston and Oancea use explicit cycles in the path space of complex projective space to compute a Pontryagin-Chas-Sullivan type product in [HO13].There they use the name completing manifold for this construction of completing relative cycles.We shall use this terminology here as well.The goal of this article is to use Ziller's completing manifolds to compute the string topology coproduct for complex and quaternionic projective space.The string topology coproduct was introduced by Goresky and Hingston in [GH09].It was furthermore studied by Hingston and Wahl in [HW17] where the authors give a definition which is equivalent to the one in [GH09] and which we shall use in this article.If M is a closed oriented Riemannian manifold of dimension N, the string topology coproduct is a map where ΛM is the free loop space of M and M is considered as a subspace of ΛM via the identification of a point with the trivial loop at this point.The string topology coproduct has been computed for odd-dimensional spheres in [HW17].Furthermore, there are partial computations of the string topology coproduct on Lens spaces, see [Nae21] and [NRW22].In particular in [NRW22] Naef, Rivera and Wahl show that the string topology coproduct is not a homotopy invariant in general.However, as Hingston and Wahl show in [HW19] if one only considers homotopy invariances with certain additional conditions then the string topology coproduct is invariant under these maps.The author of this article used Bott's K-cycles -which can be understood as completing manifolds -to show that the string topology coproduct is trivial for compact simply connected Lie groups of rank r ≥ 2, see [Ste21].In this present article we use completing manifolds to compute the string topology coproduct on M = CP n and M = HP n .We are going to explicitly describe a family of closed manifolds Γ k and embeddings f k : Γ k → ΛM, k ∈ N such that there is a family of homology classes which generate all of the homology of the pair (ΛM, M) with rational coefficients.Here [α i k ] and [β i k ] are classes in the homology of Γ k .We will then show that the string topology coproduct behaves as follows.
Theorem (Theorem 7.5).Let K be either C or H.The string topology coproduct on M = KP n satisfies We use this result to compute the Goresky-Hingston product on the manifolds CP n and HP n , see Theorem 8.1.This article is organized as follows.In Section 2 we introduce the notion of completing manifolds and discuss their relevance in Morse theory.The string topology coproduct is defined in Section 3. We study the critical manifolds of the length functional in the free loop space and in the figure eight space in Section 4. The completing manifolds are introduced in Section 5 and in Section 6 we discuss their cohomology ring.The computation of the string topology coproduct is then carried out in Section 7. Finally, in Section 8 we use the results of the previous sections to compute the Goresky-Hingston product on CP n and HP n .
The proof of a central Lemma of Section 7 is to be found in Appendix A and in Appendix B we discuss a relative version of the standard cap product which is used in the definition of the string topology coproduct.
Acknowledgements.The author thanks the anonymous referee for their careful and thoughtful reading of our manuscript.Their suggestions highly improved the exposition of this article.

COMPLETING MANIFOLDS AND MORSE THEORY
We start by introducing the notion of a completing manifold following the expositions in [HO13] and [Oan15].
Let X be a Hilbert manifold and let f : X → R be a smooth function on X satisfying the Palais-Smale condition (C).Let a be a critical value of f and assume that the set of critical points B at level a is a non-degenerate finite-dimensional critical submanifold of finite index k with orientable negative bundle.Then the behavior of the level homology H • (X ≤a , X <a ) is well known.It holds that H • (X ≤a , X <a ) ∼ = H •−k (B) where coefficients can be taken in an arbitrary commutative ring R. In applications the homology of these critical submanifolds may be much easier to understand than the homology of X.Therefore, one would like to find conditions which imply that all of the homology of X is built up by these level homologies.Definition 2.1 ([Oan15], Definition 6.1).Let X be a Hilbert manifold and let f be a smooth realvalued function on X satisfying condition (C).Let a be a critical value of f and assume that B is a non-degenerate connected critical submanifold at level a of index k and of dimension l = dim(B).Assume that k and l are both finite.A completing manifold for B is a closed, orientable manifold Γ of dimension k + l with an embedding ϕ : Γ → X ≤a such that the following holds.There is an l-dimensional submanifold L such that ϕ| L maps L homeomorphically onto B and there is a retraction map p : Γ → L. Furthermore, the embedding ϕ induces a map of pairs ϕ : (Γ, Γ L) → (X ≤a , X <a ).
(1) This definition of a completing manifold is actually the one of a strong completing manifold in [Oan15].Since all cases that we consider in this article satisfy the assumption of this stronger version we limit our attention to this situation.
(2) Note that the above definition can be used for cases where the critical set at level a consists of several connected critical submanifolds.We can then set up a completing manifold for each connected component.We will see this in the case of the figure-eight space in Section 4.
Recall that if f : M → N is a map between oriented manifolds then the Gysin map is given by Here PD B stands Poincaré duality on the manifold B. The Gysin map p ! : where s : L ֒→ Γ is the embedding of L into Γ given by the data of the completing manifold.Up to sign, the Gysin map s ! is equal to the composition where the first map is induced by the inclusion of pairs and the second is the Thom isomorphism, see [Bre13,Theorem VI.11.3].This shows that the map H See also [HO13,Remark 7].In particular this observation leads to the following result.
Proposition 2.3 ([Oan15], Lemma 6.2).Let X be a Hilbert manifold, f a smooth real-valued function on X satisfying condition (C) and a be a critical value of f .Assume that the set of critical points at level a is a non-degenerate critical submanifold B of index k.If there is a completing manifold for B then If the homology of the sublevel set X ≤a is isomorphic to the direct sum for all critical values a, we say that the function f is a perfect Morse-Bott function.This property clearly holds if all the connecting morphisms in the long exact sequence of the pair (X ≤a , X <a ) vanish.If every critical submanifold has a completing manifold, it follows that the function f is perfect.Using completing manifolds Ziller shows in [Zil77] that the energy function on the free loop space of a compact symmetric space is a perfect Morse-Bott function.Note that he uses Z 2 -coefficients in general, since there are issues with orientability for some spaces.We will describe these completing manifolds for M = CP n and M = HP n in detail in Section 4.
There is also an obvious generalization of the above Proposition if we are in the situation of the critical set decomposing into several connected components and each one admitting a completing manifold.

THE STRING TOPOLOGY COPRODUCT
In this section we introduce the string topology coproduct.We closely follow the definition of the coproduct given in [HW17].Let M be an oriented closed N-dimensional Riemannian manifold.We denote the unit interval by be the set of absolutely continuous curves in M such that their derivative is square integrable.See [Kli95, Definition 2.3.1] for the definition of absolutely continuous curves in a manifold.We define the free loop space of M to be and this is in fact a submanifold of PM.The manifold M itself can be embedded into ΛM via the trivial loops, see [Kli78,Proposition 1.4.6].On the path space PM we consider the length functional which is a continuous function on PM, see [Kli95,Theorem 2.3.20].Moreover, it is smooth on PM M. If we restrict L to the free loop space ΛM it turns out that the non-trivial critical points of L are precisely the closed geodesics in M.
We now fix a commutative ring R and consider homology and cohomology with coefficients in R. Fix an ǫ > 0 smaller than the injectivity radius of M. Then the diagonal ∆M ⊆ M × M has a tubular neighborhood given by Here, d is the distance function on M induced by the Riemannian metric.For an As Hingston and Wahl argue, see [HW17, Section 1.3], the Thom class in H N (TM, TM M) induces a Thom class τ M ∈ H N (U M , U M,≥ǫ 0 ).On the free loop space we consider the space We set We define the class We refer to [HW17, Section 1.5] for its precise definition.Finally, consider the cutting map cut : ) and reparametrizes both loops such that they are again defined on the unit interval I.Note that the cutting map actually factors through maps With this preparation we can now define the string topology coproduct.Let [I] be the positively oriented generator of H 1 (I, ∂I) with respect to the standard orientation of the unit interval.In order to shorten notation we shall also write Λ for the free loop space ΛM.
Definition 3.1.The string topology coproduct is defined as the map Remark 3.2.Let M be a closed oriented manifold.
(1) Note that the cap product with the class τ Λ is a particular relative cap product.This relative cap product is defined in Appendix B where we also study some basic properties.
(2) Hingston and Wahl define an algebraic loop coproduct, see [HW17, Definition 1.6], which is a sign-corrected version of the string topology coproduct.Since we will later only consider even-dimensional manifolds, this sign correction does not matter.
If we use field coefficients, then the string topology coproduct induces a dual product in cohomology which is known as the Goresky-Hingston product.
Definition 3.3.Let F be a field and assume that the homology of ΛM is of finite type.Let α ∈ H i (Λ, M; F) and β ∈ H j (Λ, M; F) be relative cohomology classes, then the Goresky-Hingston product α * β is defined to be the unique cohomology class in H i+j+N−1 (Λ, M; F) such that Remark 3.4.Let M be a closed oriented manifold.
(1) The Goresky-Hingston product can also be defined intrinsically, see [GH09].However, in this article we shall study properties of this product only via the duality with the string topology coproduct.(2) As for the string topology coproduct, Hingston and Wahl define a sign-corrected version of the Goresky-Hingston product in [HW17].For even-dimensional manifolds, the above product and its sign-corrected version agree.Hence, in this article the distinction will not matter.
Remark 3.5.Since the next four sections will deal with the technical details of the computation of the coproduct we want to sum up the strategy for computing the string topology coproduct on M = CP n or M = HP n at this point.
• In Section 4 we will study the critical manifolds Σ k , k ∈ N in ΛM of the length functional L : ΛM → R. • In Section 5 we construct the completing manifolds Γ k .
• We shall see that the manifold Γ k , k ∈ N can also serve as a completing manifold for critical submanifolds in the figure-eight space ΛM × M ΛM. • We determine the cohomology ring of Γ k in Section 6.We can then explicitly compute the Gysin map and obtain a set of generators for the homology H • (ΛM, M). • In Section 7 we will then replicate all the steps in the definition of the coproduct on the manifold Γ k .• We will pull back the class τ Λ to a class which can be described in terms of the cohomology of Γ k and compute the cap product with this class.• Then one sees that under the cutting map cut : F Λ → ΛM × M ΛM we get homology classes which we can identify with classes coming from the manifold Γ k seen as a completing manifold in the figure-eight space ΛM × M ΛM.

CRITICAL MANIFOLDS IN THE FREE LOOP SPACE OF PROJECTIVE SPACES
In this section we describe the completing manifolds on the loop space of CP n and HP n in detail.We will first study the critical manifolds of ΛM and ΛM × M ΛM with respect to the length functional L. Then we define the completing manifolds Γ k , k ∈ N and show that Γ k can serve as a completing manifold both in ΛM as well as in ΛM × M ΛM.Finally, we describe the cohomology ring of Γ k in detail and give a set of explicit generators of H • (ΛM, M).
From now on let K = C or K = H.In case K = C we set N = 2n, and λ = 1 and in case K = H we set N = 4n, and λ = 3.We consider the free loop space of M = KP n where we consider M as a symmetric space.The symmetric Riemannian metric on M induces a length functional L : ΛM → R, see equation (3.1) and it is well-known that this is a Morse-Bott function on ΛM.Moreover, the index and the nullity of all critical manifolds are finite, see [Zil77].There is a group G with a closed subgroup K such that M = G/K and (G, K) is a Riemannian symmetric pair.In particular the action of G is a transitive action by isometries and K is the isotropy group of a fixed basepoint p 0 ∈ M. The group K then acts on M by isometries as well and fixes the basepoint.It is well-known that all geodesics on M are closed and of the same prime length l.Consequently, the critical values of the length functional are positive multiples of l.If k ∈ N then the critical set at level a = kl is diffeomorphic to the unit tangent bundle SM, i.e.
is a diffemorphism.Moreover, Ziller argues in [Zil77] that the group G acts transitively on Σ a and this action equals the canonical action of G on the unit tangent bundle.Let γ ∈ Σ a be a closed geodesic at level a = kl with γ(0) = p 0 .Then there is an underlying prime geodesic σ ∈ Σ l such that γ = σ k .Denote the isotropy group of γ with respect to the action of G on Σ a by K γ .In particular, this is a closed subgroup of K. Then we have Furthermore, there is an induced action of the group K on Σ a and the orbit of γ is which is the fiber of SM over p 0 .In the following we will always write Σ k for the critical submanifold at level kl instead of see [GH09,p. 167] and the nullity is equal to the dimension of Σ k , i.e.

null(Σ
On the figure-eight space ΛM × M ΛM we consider the length function for (η 1 , η 2 ) ∈ ΛM × M ΛM.The critical manifolds in ΛM × M ΛM are the sets of the form and the critical values are again multiples of l.Here the fiber product is taken with respect to the evaluation map at time t = 0, i.e.
At level kl the critical set is Note that by the relative construction of the coproduct the components M × M Σ k and Σ k × M M will not show up in the course of the proof so we will not deal with them.See also Remark 7.6.
Proof.The function clearly satisfies the Palais-Smale condition.Since ΛM × M ΛM is a closed submanifold of Λ × Λ and since L 2 is the restriction of L it therefore follows that L 2 also satisfies the Palais-Smale condition.
In order to show that L 2 is a Morse-Bott function, we need to show the following property.Let (η 1 , η 2 ) ∈ ΛM × M ΛM be a critical point of L 2 and assume that it belong to the critical submanifold of the form Σ a × M Σ b where Σ a and Σ b are critical submanifolds in ΛM with respect to the Morse-Bott function L. We need to show that the null space Section 2] that all periodic Jacobi fields along closed geodesics in a compact symmetric space are restrictions of Killing vector fields.Let ξ 1 , ξ 2 be periodic Jacobi fields along η 1 and η 2 , respectively.We assume without loss of generality that η 1 (0) = η 2 (0) = p 0 is the basepoint.If ξ 1 (0) = ξ 2 (0) = 0 then both Jacobi fields are restrictions of Killing fields on M which are induced by the action of the group K. Since the action of In this case one sees as in [Zil77, page 8] that both Jacobi fields are restrictions of the same Killing field, since the Killing field are determined by the element ξ 1 (0) in this case.Hence these Jacobi fields can be understood as restrictions of a Killing field of the diagonal group action G × M × M → M × M. Therefore in this case we also see that (ξ 1 , ξ 2 ) ∈ T (η 1 ,η 2 ) Σ a × M Σ b .Since the Jacobi fields of the above two types form a basis of T 0 (η 1 ,η 2 ) Λ × M Λ this shows the inclusion Consequently, L 2 is a Morse-Bott function and the claim for the nullity then follows directly from the dimensions of the critical submanifolds.Finally, for the indices, note that the index of a closed geodesic in a compact symmetric space is the same whether we consider it as a critical point in the based loop space or in the free loop space.Therefore, we get and thus we see that the inequalities are all equalities.This completes the proof.
For i, j ≥ 1 such that i + j = k we have where pr : SM → M is the canonical projection of the unit sphere bundle of M.Moreover, the projection onto the first factor makes SM × M SM into a sphere bundle over SM which admits a global section Since later on we shall use the cohomology ring of SM × M SM, we prove the following Lemma.
Lemma 4.2.The rational cohomology ring of SM × M SM is isomorphic to where deg(α Proof.From the Gysin sequence for SM → M we know that The manifold SM × M SM is the total space of a sphere bundle over SM with a global section, therefore it follows from the corresponding Gysin sequence that with a generator ξ of degree deg(ξ) = N − 1.This proves the claim.
For i ∈ {0, . . ., n − 1} we denote the homology class dual to the class α i by [a i ] and the dual of Before we turn to the completing manifolds, let us note a property of the conjugate points along the closed geodesics in M. With γ and σ as before, note that there is precisely one conjugate Moreover, the index of σ is equal to λ, since the index of a closed geodesic on a compact symmetric space is equal to the sum of the multiplicity of the interior conjugate points, see again [Bes78, Proposition 3.35] and [Zil77].Denote the isotropy group of this point with respect to the action of K by K a .It is well-known that dim(K a ) > dim(K γ ), see [Zil77, Theorem 4], and that dim(K a ) − dim(K γ ) is equal to the index of σ both as a geodesic loop in ΩM as well as a closed geodesic in ΛM.
Lemma 4.3.The homogeneous space K a /K γ is diffeomorphic to the sphere S λ .
Proof.By [Bes78, Proposition 3.35] the set of first conjugate points along geodesics of the basepoint p 0 is equal to the cut locus of p 0 .Moreover, we have is well-known to be the round sphere S N−1 .Moreover, the exponential map induces a fiber bundle exp p 0 : S N−1 → KP n−1 , see [Bes78, Proposition 3.37].These are just the well-known fibrations of spheres over projective space, so it follows that the fiber is S λ .See also [Bes78,Theorem 5.29] for details.We can understand these objects as homogeneous spaces, i.e. as we know from equation (4.1) we have K/K γ ∼ = S N−1 and it is clear that K/K a is diffeomorphic to the cut locus KP n−1 .Therefore we see that that the fiber K a /K γ of the fiber bundle K/K γ → K/K a is diffeomorphic to S λ .

CONSTRUCTION OF ZILLER'S COMPLETING MANIFOLDS
In this section we describe Ziller's completing manifolds, see [Zil77].We describe the manifolds and the respective embeddings in detail.Fix a closed geodesic γ = σ k ∈ ΛM of multiplicity k starting at the basepoint p 0 ∈ M. Here, σ is the underlying prime closed geodesic.Consider the product with 2k factors in total.Throughout this section, we will follow the convention that the first element in the tuple (g 0 , x 1 , . . ., is said to be at zero'th position, the second one at first position and so forth.The element in zero'th position plays a special role since it lies in G, therefore we will denote it usually by g 0 while the other elements will be denoted by x i .There is a right action of the 2k-fold product of K γ on W k given by ).This action is free and proper and we consider the quotient space Note that the critical submanifold Σ k ∼ = G/K γ can be seen as a submanifold of Γ k via the em- bedding s L,k : We define L k = s L,k (G/K γ ) and will identify L k and G/K γ in the following possibly without making the identification explicit.There is a submersion Hence, the map Note that the only closed geodesics in the image of f k are precisely the closed geodesics in the critical submanifold Σ k .All other loops in im( f k ) are broken geodesics and hence they are not critical points of the energy functional.Therefore, the flow of the energy functional decreases the value of L for all γ ∈ im( f k ) which are not in Σ k .Consequently, if we compose the embedding f k with an arbitrary short gradient flow of the length functional, we obtain a map of pairs we have shown that Γ k is a completing manifold for Σ k if we prove that it is orientable.We will see the orientability later.Remarkably, Γ k can also serve as a completing manifold in the Hilbert manifold ΛM × M ΛM.
With k and γ as above, fix 1 ≤ m ≤ k − 1 and consider the critical point where σ is the underlying prime geodesic of γ.The component of the critical set at level k in Like the action χ above, this is a free and proper right action and we consider the quotient space V = G × K/χ ′ .
Lemma 5.1.The manifold V is diffeomorphic to SM × M SM.
Proof.Recall that there is a diffeomorphism G/K γ → SM induced by the transitive action of G on SM.In particular we see that These maps factor through the submersions G × K → V and E → SM × M SM and therefore induce smooth maps It is a direct computation that they are inverses of each other.
Observe that there is an embedding s V,m : V ֒→ Γ k given by s V,m ([g 0 , x 2m ]) = [g 0 , e, . . ., e, x 2m , e, . . ., e] ∈ Γ k where x 2m appears at the 2m'th position.We denote the image of V under this embedding by V.There is a submersion p V,m : where Proof.We show that the diagram With this identification the commutativity of the above diagram can be checked using the respective definitions.The second statement can be checked from the definition of the map F k,m .
If we compose F k,m with an arbitrarily short flow of the gradient flow of the length functional L 2 on ΛM × M ΛM we obtain a map of pairs Note that in ΛM × M ΛM we can also use the fiber product Γ m × M Γ k−m as a completing manifold where the fiber product is taken with respect to the evaluation map as an embedding of Γ m × M Γ k−m and can check that this is again a completing manifold.To conclude this section we want to show that the completing manifolds for [g 0 , x 1 , . . ., x 2k−1 ] ∈ Γ k .Note that this is a well-defined and smooth map since it descends from an equivariant map One checks again that Ψ m is well-defined and smooth.
Here, p L,m : Γ m → SM and p L,k−m : Γ k−m → SM are the submersions which are used for the completing manifold structure in the free loop space, see equation (5.1).In particular Φ is a diffeomorphism.
Proof.This can be checked directly by unwinding the definitions.

COHOMOLOGY OF THE COMPLETING MANIFOLDS
The manifold Γ k is closely related to the K-cycles in the sense of Bott and Samelson, see [BS58].In this section we shall describe its homology and cohomology following the discussions of the cohomology of the K-cycles by Bott and Samelson [BS58] and by Araki [Ara62].Recall that Γ k is defined as the quotient of W = (G × K a ) × (K × K a ) k−1 modulo the action of K 2k γ via χ((g 0 , x 1 , . . . ,x 2k−1 ), (h 0 , . . ., ).If we have any product of subgroups K i ⊆ G, i ∈ {1, . . ., m} such that K γ ⊆ K i for all i ∈ {1, . . ., m} there is an action of (K γ ) m on this product given by We denote the quotient by Note that we might have that the first group K 1 is the group G.All other groups will be subgroups of K. Lemma 6.1.Let 0 ≤ i 1 < i 2 < . . .< i m ≤ 2k − 1 be integers and set K 0 = G, K j = K for j even, j ≥ 2 and K j = K a for j odd.Then the manifold Proof.We define a map σ i 1 ...i m : = (e, . . ., e, x i 1 , e, . . ., e, x i 2 , e, . . . ,e, x i m , e . . . ,e) where x i j is at position i j for each j ∈ {1, . . ., m}.If = (e, . . ., e, x i 1 k 1 , e, . . ., e, k −1 Hence, σ i 1 ...i m is equivariant with respect to the action of K m γ on K i 1 × . . .× K i m and the action χ of K 2k γ on W k .Therefore this yields a smooth map It is easy to check that this is an embedding. From now on we will always identify a sub-Kcycle Γ P with its image in Γ k under the embeddings constructed in Lemma 6.1.The manifold V ∼ = V which we defined above is an example of a sub-K-cycle.If P = (0, 1, . . ., m) for some m ≤ 2k − 2, then there are submersions given by p Γ,m ([g 0 , x 1 , . . ., Hence, we get a chain of submersions As shown in [Ara62, Theorem 2.4] all these submersions are fiber bundles.Moreover, each fiber bundle has a section which is given by the map be a sequence of manifolds with each π i : E i → E i+1 being a sphere bundle.Then we say that E 0 is an iterated sphere bundle over B.
Proposition 6.3.The K-cycle Γ k is an iterated sphere bundle via the maps Proof.The fibers of the iterated fiber bundle are either K/K γ of K a /K γ .We know that K/K γ ∼ = S N−1 and K a /K γ ∼ = S λ , see Lemma 4.3.Hence, Γ k is an iterated sphere bundle over SM ∼ = G/K γ .
In the situation of an iterated sphere bundle E 0 → E 1 → . . .→ E m = B, one can compute the cohomology of the total space E 0 by considering the Gysin sequences at each step.In the following we consider homology and cohomology with rational coefficients.We determine the cohomology ring of Γ k .Note that throughout the article we have fixed an orientation on M. In particular this induces an orientation on the ǫ-sphere around the basepoint p 0 .We shall denote the generator of its fundamental class by [S N−1 ǫ ].Furthermore, if t ∈ I, define ev t : ΛM → M to be the map ev t (γ) = γ(t) for γ ∈ ΛM.Proposition 6.4.The cohomology ring of Γ k is isomorphic to In particular Γ k is orientable.Furthermore, the class ξ 2i and the dual class [x 2i ] in homology for i ∈ {1, . . ., k − 1} can be chosen such that the following holds.For i ∈ {1, . . ., k − 1} one can choose a fundamental class for some small δ > 0 and where s 2i : S N−1 → Γ k is the embedding constructed in Lemma 6.1.
Proof.As we have seen before it follows from the Gysin sequence of SM → M that with deg(α) = λ + 1 and deg(β) = N + λ.One can now determine the cohomology ring of Γ k by induction along the steps of the iterated sphere bundle.Note that for a k-sphere bundle E → B with k odd and which admits a global section one has where deg(ξ) = k.Thus one iteratively obtains the cohomology ring.For the orientations we note that at each step in the iterated sphere bundle we are free to choose the orientation of the new generator ξ i .As Araki argues using Gysin sequences, see [Ara62, Section 2], the homology class dual to the class ξ l for l ∈ {1, . . ., 2k − 1} can be described as follows.Let s l : K l /K γ ֒→ Γ k be the embedding as in Lemma 6.1.Then if we consider an orientation class with δ > 0 small then it can be seen directly that the map ev for some small ǫ ′ > 0. Note that this property holds precisely because all geodesics in M are closed and of the same prime length.We can now choose an orientation class [S N−1 ] and correspondingly the class ξ 2i and its dual [x 2i ] such that In the previous proposition we saw that the manifolds Γ k , k ∈ N are orientable.This completes the proof that the Γ k are in fact completing manifolds.We sum this up in the next corollary.Corollary 6.5.Let M = KP n be a complex or quaternionic projective space and let k, m ∈ N with k > m.
(1) The manifold Γ k with the embedding f k is a completing manifold for the critical set Σ k in ΛM.
(2) The manifold Γ k with the embedding F k,m is a completing manifold for the critical set Note that the Corollary implies the perfectness of the Morse-Bott function L on ΛM.This is one of the main results in [Zil77].
Recall that at level kl the level homology H • (ΛM ≤kl , ΛM <kl ) is isomorphic to the homology of the critical submanifold L k ∼ = SM.Moreover, as we have seen in Section 2 the map . Hence, we obtain the generators of H • (ΛM) which come from level k by considering the map (p L,k ) ! .Recall that L k ∼ = SM and as we have seen its cohomology ring is In particular, we can choose these generators such that under the embedding 2) where p L,k : Γ k → L k is the retraction.The above formulas also justify the misuse of notation, since e.g. the cohomology class α has a double meaning, but as (p L,k ) * : H Recall that the Gysin map (p L,k ) ! : H Using Proposition 6.4 we can now compute the map (p L,k ) ! .
Lemma 6.6.With the above notation the following equations hold Proof.We just consider the first case, the second case is analogous.The Poincaré dual of . By equation (6.2) we have We now need to compute the Poincaré dual X = PD Γ k (α n−1−i β) of this latter class.We compute the Kronecker pairing Now, since α is of even degree and β is of odd degree we get by graded commutativity We define classes +N−1 (ΛM) for k ∈ N and i ∈ {0, . . ., n − 1}.Note that the degree of all A i k is odd, while the degree of all B i k is even.The following is then clear by the construction of the completing manifolds.Proposition 6.7.The homology of the free loop space relative to the constant loops is generated by the image of the set

COMPUTATION OF THE COPRODUCT
In this section we explicitly compute the string topology coproduct.As we have seen in the previous section we can explicitly describe a set of generators of the homology H • (ΛM, M) via the completing manifolds Γ k , k ∈ N. We therefore express all the steps in the definition of the coproduct in intrinsic terms of the manifolds Γ k .First, we want to pull back the class τ Λ ∈ H N (U Λ , U Λ,≥ǫ 0 ) via f k to a class which can be described in terms of the cohomology of Γ k .We consider the preimage ( f k , id I ) −1 (U Λ ) ⊆ Γ k × I.This set can be described explicitly as follows.Since all loops in the image f k (Γ k ) are broken geodesics, there is a small δ > 0 such that This is because every other conjugate point on a closed geodesic starting at the basepoint is the basepoint itself.Clearly, δ > 0 is so small that the open intervals are disjoint.Similarly, there is a δ 0 > 0 with δ 0 < δ such that . To make the bookkeeping easier, let us define have trivial homology for obvious reasons.For m ∈ {1, . . ., k − 1} we have We choose positively oriented generators [I m ] of H 1 (I m , J m ) and dual cohomology classes We now want express the cohomology class is given by This key lemma is proved in Appendix A. In the following let Y be one of the classes where X ∈ H • (Γ k ) is the respective homology class in Γ k .In order to compute ∨Y we need to consider by naturality of the cap product, see Proposition B.1.As seen in Lemma 7.1, we obtain By the compatibility of the cross and the cap product, see Proposition B.2, we have By the construction of the relative cap product we see that Lemma 7.2.The relative cap product yields Proof.Using Proposition 6.4 we have ] and this yields the claim.
For convenience of notation we shall write and plug in the respective classes later using the above Lemma.Then we have Fix an m ∈ {1, . . ., k − 1}.To finish the computation of ∨Y, we need to determine the effect of the retraction map R GH and of the cutting map on commutes, where σ m : I m → { 2m 2k } is the constant map.To complete the computation, we need to characterize A direct computation shows that the map which was defined in equation (5.3), up to the obvious identification Here and in the following [ 2m 2k ] ∈ H 0 ({ 2m 2k }) denotes the canonical generator.We now want to express the class (ι • F k,m ) * X m ∈ H • (ΛM × ΛM) as a product of the generators A i k and B i k .Here, ι : ΛM × M ΛM ֒→ ΛM × ΛM is the inclusion of the figure-eight space.In order to do so we need the following lemma.

Lemma 7.3. The following diagram commutes
where the vertical arrows in the lower row are induced by the respective inclusions.
Proof.The commutativity of all subdiagrams is clear apart from the lower left square.In order to show that the lower left square commutes, we first consider the diagram

Th
where Th : Here, U is a tubular neighborhood of is the Thom isomorphism.The map Th ′ is defined analogously.In particular, we note that the normal bundle of Therefore the above diagram commutes.Now, note that the map Th agrees with the Gysin map (s m , s k−m ) ! .This follows from [Bre13, Theorem VI.11.3].Note that in this reference it is only claimed that the two maps agree up to sign, but one can determine the sign from the proof.Applied to our present case the sign is (−1) c(λ k −(N−1)) for some integer c ∈ Z. Recall that the index λ k is odd for all k ∈ N. Consequently, the codimension is even for all k ∈ N, so we see that the sign (−1) c(λ k −(N−1)) is even and thus the maps Th and (s L,m , s L,k−m ) ! agree.Now, let Z ∈ H • (SM × M SM).Then we have Moreover, let us denote the inclusion SM × M SM ֒→ SM × SM by i 1 and denote the inclusion and this shows the commutativity of the lower left square.
Recall that Γ k together with the embedding F k,m is a completing manifold for the critical set In Lemma 4.2 we determined the cohomology ring of V. We make the following choice of orientation.The generator ξ ∈ H N−1 (V) is chosen in such a way that it pulls back to the generator ξ 2m under the map p V,m .Note that by the choice of orientations for the classes ξ 2i , i ∈ {1, . . ., k − 1} in Proposition 6.4 this is a consistent choice.We choose the orientation for V such that α n−1 βξ is a fundamental cohomology class.
Lemma 7.4.Using the notation for the generators of H • (SM × M SM) as in the paragraph following Lemma 4.2 we have Proof.We only consider the first case, the second one is analogous.The Poincaré dual of Moreover, by our orientation convention, the pullback of this cohomology class is Now we need to compute the Poincaré dual of this class in Γ k , i.e.
By using the graded commutativity of the cup product, we see that Therefore we see that By the above Lemma we see that we can write every class which shows up in Lemma 7.2 as a class in the image of (p V,m ) ! .Thus, the commutative diagram in Lemma 7.3 enables us to express the classes in the coproduct through the generators A i k and B i k .Thus we need to understand the effect of the map on the classes [a i ] and [a i b].Note that these classes can be described as push-forward of the classes [a i ] and [a i b] in H • (SM) under the embedding SM ֒→ V given by where d : SM → SM × SM is the diagonal map.Via the cup ring of SM, it is easy to figure out the effect of the diagonal map in homology.We have We obtain the following final result.
Theorem 7.5.Let K be C or H and consider M = KP n .The string topology coproduct on M behaves as follows.We have ).By Lemmas 7.1 and 7.2, we have Then by Lemmas 7.3 and 7.4 we see that Hence we are left with computing the Gysin map of the retraction (p L,m , p L,k−m ).By [Bre13, Proposition VI.14.3] we see that is odd for all m ∈ N and that dim(Γ k−m ) is even for all m, k ∈ N, m < k we can figure out the sign.The computation of ∨B i k is analogous.Remark 7.6.Note at this point that it is sufficient to consider the critical manifolds with respect to the length function L 2 of the form There are two other connected components at level kl, namely M × M Σ k and Σ k × M.However, as we saw now the corresponding homology classes in H • (ΛM × M ΛM) do not show up in the process of computing the coproduct, therefore we did not consider these components.

THE COHOMOLOGY PRODUCT
In this section we briefly describe the Goresky-Hingston product on KP n .We take cohomology with rational coefficients.Define classes k is defined to be the dual of A i k and the class µ i k is defined to be the dual of . ., n − 1}} generates the cohomology H • (ΛM, M) additively.Using Theorem 7.5 we see that whose products are subject to the above relations.In particular, the ring is finitely generated and the element σ 0 1 is non-nilpotent.Remark 8.2.The existence of a non-nilpotent element is already shown in [GH09, Theorem 14.2].The fact that the Goresky-Hingston ring is finitely generated is analogous to behaviour of the Goresky-Hingston product on spheres, see [GH09].APPENDIX A. PROOF OF LEMMA 7.1 In this appendix we prove Lemma 7.1.We will use the notation established throughout the paper.We shall show that the pullback of the class τ Λ ∈ H N (U Λ , U Λ,≥ǫ 0 ) under the map First, let us consider the generators of H N−1 (Γ k ).The classes ξ 2 , . . ., ξ 2k−2 are generators of H N−1 (Γ k ), so are the classes where necessarily m(λ + 1) + l λ = N − 1.One can check that these are the only generators.Hence, we have where λ j,m ∈ Q and ρ m,i 1 ,...,i l ,s ∈ Q are coefficients and where the sum in the second term is taken over those combinations of m, i 1 , . . ., i l such that m(λ + 1) + l λ = N − 1 and all i j odd.
Here, the x appears at position 2j.Then = (p 0 , q) for q ∈ B p 0 is the inclusion of the fiber of the normal tubular neighborhood then we have is a positively oriented generator.We define the map Note that this factors through maps Define ev : Λ × I → M by ev(γ, s) = γ(s).In order to show that it thus suffices to shows that Recall that the orbit of a point q ∈ B p 0 under K is the distance-sphere around p 0 of radius d(p 0 , q).We compute the map h ′ m explicitly.Let x ∈ K.In case that m < j, we have and in case m > j, we obtain x.γ(t) t ≥ 2m 2k .Now, consider the following commutative diagram where the maps ∂ are the respective connecting homomoprhisms.The middle square is id with h 1 m,j and h 2 m,j the maps induced by (h ′ m ) * .In case m < j, it is clear that h 2 m,j is the trivial map, so h 1 m,j = 0.This can be seen from equation (A.3) since the map h ′ m is homotopic to a locally constant map.If m > j, we have by the orientation convention of Proposition 6.4 that h 2 m,j (x, y) = x + y, so again h 1 m,j is trivial.Finally, for m = j, we see that h 2 m,m (x, y) = y, so we get h 1 m,m = id .Here we use again the orientation convention established in Proposition 6.4.This shows that The claim then follows from equations (A.1) and (A.2).
Proof.Fix m ∈ {0, . . ., n − 1} and odd integers i 1 , . . ., i l with 1 ≤ i 1 < . . .< i l ≤ 2k − 1 such that m (λ + 1) + l λ = N − 1.We also fix s ∈ {1, . . ., k − 1}.We begin by describing a dual class to the cohomology class Let κ = m(λ + 1).Recall that π L : L k ∼ = SM → M is the unit sphere bundle of the underlying manifold.Since κ < N we see from the Gysin sequence of this sphere bundle that there is an isomorphism We know that H κ (M) ∼ = Q.Note that a generator of H κ (M) can be described as follows.It is well-known that there is an inclusion of KP m into M = KP n which maps a fundamental class of KP m to a generator of H κ (M).In particular we can choose this inclusion in such a way that the basepoint p 0 ∈ M is also the basepoint of KP m .Denote this inclusion by j : KP m → KP n .We pull back the unit sphere bundle along j.Since the dimension of the fiber of this bundle is greater than the dimension of the base the Euler class vanishes and hence this bundle has a section s : KP m → j * SM.We compose this with the canonical map j * SM → SM to get a section σ : KP m → SM.It is clear that we have π L • σ = j, so we see that we obtain an isomorphism Hence, we can represent a generator in degree κ by the image of a fundamental class of KP m under the map σ.Consider the manifold Γ 0i 1 ...i l where we use the notation of Lemma 6.1.We have In Lemma 6.1 we saw that this embeds into the manifold Γ k via a map s 0i 1 ...i l : Γ 0i 1 ...i l → Γ k .Moreover, as in the discussion after the proof of Lemma 6.1 one sees that this manifold is an iterated sphere bundle over SM and the fiber at each step of the iterated sphere bundle is the sphere S λ .We now pull back this fiber bundle along the embedding σ : KP m ֒→ SM to get a pull-back space X which is clearly a manifold.We have a commutative diagram where the left square is just the pullback diagram.The manifold X is also an iterated sphere bundle with base KP m .Moreover, from the Gysin sequences of X, Γ 0i 1 ...i l and Γ k one can see that (ι m ) * [X] is indeed the dual homology class to the cohomology class α m ∪ ξ i 1 ∪ . . .∪ ξ i l .Consequently, we obtain where we have g = ev Λ • ( f k , id I s ) as before.We define vanishes, then by equation (A.4) we have shown that the coefficients ρ m,i 1 ,...,i l ,s vanish.Let us compute the map h m,s explicitly.There is a number o ∈ {1, . . ., l} such that We see from the above expression that the map h m,s factors as follows where t : KP m ֒→ KP n = M is the inclusion.We now consider the following commutative diagram Note that the space t * U M is homeomorphic to a disk bundle over KP m and is therefore homotopy equivalent to KP m itself.Clearly, the dimension of KP m satisfies dim(KP m ) ≤ N − 2 and therefore the homology groups on the very left and the very right in the lower row of the above diagram vanish.Therefore the connecting homomorphism in the lower row is an isomorphism.Moreover, note that J s is adisjoint union of two intervals, hence it is homotopy equivalent to a union of two points, i.e. we have a homotopy equivalence We thus consider the maps Note that X ∼ = X × {t ± }, so we can understand both maps as maps X → t * U M,≥ǫ 0 .We now show that they induce the same map in homology.First, we define a map ϕ : X → X.Note that since M is a symmetric space there is an isometry S : M → M which fixes the basepoint p 0 = γ(0) and acts as −id T p 0 M on the tangent space.This isometry reverses geodesics going through the basepoint.Since the point a ∈ M is the unique conjugate point in the interior of the prime closed geodesic σ we have S.a = S.γ( 12 ) = γ(− 1 2 ) = a.Consequently, the isometry satisfies S ∈ K a .Therefore, we can define a map Note that the element S commutes with all elements in K.This is because the isotropy representation of a symmetric space is faithful and the element −id ∈ O(N) is clearly in the center of O(N).Hence, one sees that the map Φ is indeed equivariant with respect to the (K γ ) l+1 -action and therefore induces a smooth map ϕ ′ : Γ 0i 1 ...i l → Γ 0i 1 ...i l .Since this map squares to the identity it is a diffeomorphism.Moreover it clearly respects the fiber bundle structure, so it restricts to a map ϕ : X → X.We consider this map for the following reason.Since the isometry S reverses geodesics through the basepoint we have Hence, if we show that ϕ * : H κ (X) → H κ (X) is the identity, it follows that k m,s,+ and k m,s,− induce the same map in homology.We shall argue that the degree of ϕ is 1.Since ϕ is a diffeomorphism it suffices to check whether the differential at a given point is orientationpreserving or orientation-reversing.Let [e, e, . . ., e] ∈ Γ 0i 1 ...i l .There is an open neighborhood of [e, . . ., e] ∈ Γ 0i 1 ...i l−1 such that the fiber bundle p : Γ 0i 1 ...i l → Γ 0i 1 ...i l−1 is trivial over U, i.e.
But in this local trivialization it is very easy to understand the effect of the map ϕ.We have ϕ| p −1 (U) : p −1 (U) → p −1 (U) satisfies ϕ(u, x) = (u, −x) for u ∈ U, x ∈ S λ .Now, the identity on U is clearly orientation-preserving as is the antipodal map on an odd-dimensional sphere.Therefore we get that ϕ * = id H κ (X) .Now, consider again the class It is well-known that the connecting homomorphism maps this to But as we have seen now

APPENDIX B. RELATIVE CAP PRODUCT
In this section we review the construction of the relative cap product which is used in the definition of the string topology coproduct.We closely follow [HW17, Appendix A].Assume that X is a topological space with subspaces A, B ⊆ X such that is a quasi-isomorphism.Then there is a relative cap-product ∩ : H k (X, A) ⊗ H m (X, A ∪ B) → H m−k (X, B). (B.1) The condition on the subspaces is satisfied if e.g. both A and B are open.See [Bre13, Section VI.5] for details.Assume now that U 0 ⊆ U 1 ⊆ X are open subsets of X such that U = {U 1 , int(U c 0 )} is an open cover of X.Here we use the notation U c 0 = X U 0 .Furthermore, assume that A ⊆ X is another subset which is not necessarily required to be a subset of U 0 or U 1 .We assume that the intersections U 1 ∩ U c 0 and U 1 ∩ A are such that the relative cap-product 0 ), we define a map t ∩ ′ : C m (X, A) → C m−k (U 1 , U 1 ∩ A) as the composition of the maps where ρ is a map that subdivides chains with respect to the open cover U , e.g.barycentric subdivision, see e.g.[Bre13, Section IV.17].The cap product in the last step is then a chain-level version of the cap product in equation (B.1).This composition is a chain map and therefore induces a map in homology which we will refer to as cap product as well.From now on, we will also denote it by ∩ and from the context it will be clear whether we are referring to this cap product or to the one of equation (B.1).We now state a naturality statement for the cap product.Proposition B.1.Let (X, U 1 , U 0 ) and (Y, V 1 , V 0 ) be triples of spaces and let A ⊆ X and B ⊆ Y be subsets such that the inclusion is a quasi-isomorphism and similarly for V 1 , V 0 and B. Then if f : (X, Proof.The proof can be done analogously to the one of [HW17, Lemma A.1] since the restriction A ⊆ U 0 in the proof of [HW17, Lemma A.1] is not necessary.
We also need a statement about the compatibility of the relative cap product and the usual cross products.
Proposition B.2.Let X and Y be topological spaces.Furthermore, let A, U 0 , U 1 ⊆ X be subspaces such that (X, U 1 , U 0 ) is a triple of spaces and such that the inclusion ) is a quasi-isomorphism then clearly the same property holds for ).Consequently, the proof of the analogous property for the usual cap product carries over, see [Bre13,Theorem VI.5.4].
Example B.3.Take X = I to be the unit interval with A = ∂I = {0, 1}.Furthermore, choose a small δ > 0 and a number δ 0 > 0 with δ 0 < δ.Clearly, the homology group H 1 (I, ∂I) is generated by a class [I] which is represented by the relative cycle σ : I → I, σ = id I .
We use this example in Section 7.
its image under this injection.We choose the orientation of L k by choosing the class [a n−1 b] ∈ H 2N−1 (L k ) as fundamental class for all k ∈ N and as fundamental class for Γ k we choose the class [a n−1 bx 1.