1 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 [18] 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 [3] and [4] as well as Bott’s and Samelson’s work in [6]. Hingston and Oancea use explicit cycles in the path space of complex projective space to compute a Pontryagin-Chas-Sullivan type product in [8]. 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 [7]. It was furthermore studied by Hingston and Wahl in [9] where the authors give a definition which is equivalent to the one in [7] 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

$$\begin{aligned} \vee :\textrm{H}_i(\Lambda M,M)\rightarrow \textrm{H}_{i+1-N}(\Lambda M\times \Lambda M,\Lambda M\times M\cup M\times \Lambda M) \end{aligned}$$

where \(\Lambda M\) is the free loop space of M and M is considered as a subspace of \(\Lambda M\) 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 [9]. Furthermore, there are partial computations of the string topology coproduct on Lens spaces, see [14] and [15]. In particular in [15] Naef, Rivera and Wahl show that the string topology coproduct is not a homotopy invariant in general. However, as Hingston and Wahl show in [10] 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 \(r\ge 2\), see [17]. In this present article we use completing manifolds to compute the string topology coproduct on \(M =\mathbb {C}P^n\) and \(M =\mathbb {H}P^n\). We are going to explicitly describe a family of closed manifolds \(\Gamma _k\) and embeddings \(f_k:\Gamma _k\rightarrow \Lambda M\), \(k\in \mathbb {N}\) such that there is a family of homology classes

$$\begin{aligned} A_k^i = (f_k)_*[\alpha _k^i] \in \textrm{H}_{\bullet }(\Lambda M,M;\mathbb {Q}) \quad \text {and}\quad B_k^i =(f_k)_*[\beta _k^i]\in \textrm{H}_{\bullet }(\Lambda M,M;\mathbb {Q}), \end{aligned}$$

\( k\in \mathbb {N},i\in \{0,\ldots ,n-1\} \) which generate all of the homology of the pair \((\Lambda M,M)\) with rational coefficients. Here \([\alpha _k^i]\) and \([\beta _k^i]\) are classes in the homology of \(\Gamma _k\). We will then show that the string topology coproduct behaves as follows.

Theorem

(Theorem 7.5) Let \(\mathbb {K}\) be either \(\mathbb {C}\) or \(\mathbb {H}\). The string topology coproduct on \(M = \mathbb {K}P^n\) satisfies

$$\begin{aligned} \vee A_k^i = \sum _{m=1}^{k-1} \sum _{j=0}^i A_m^j\times A_{k-m}^{i-j} \end{aligned}$$

and

$$\begin{aligned} \vee B_k^i = \sum _{m=1}^{k-1} \sum _{j=0}^i ( B_m^j \times A_{k-m}^{i-j} - A_m^j \times B_{k-m}^{i-j} ). \end{aligned}$$

We use this result to compute the Goresky-Hingston product on the manifolds \(\mathbb {C}P^n\) and \(\mathbb {H}P^n\), see Theorem 8.1.

This article is organized as follows. In Sect. 2 we introduce the notion of completing manifolds and discuss their relevance in Morse theory. The string topology coproduct is defined in Sect. 3. We study the critical manifolds of the length functional in the free loop space and in the figure eight space in Sect. 4. The completing manifolds are introduced in Sect. 5 and in Sect. 6 we discuss their cohomology ring. The computation of the string topology coproduct is then carried out in Sect. 7. Finally, in Sect. 8 we use the results of the previous sections to compute the Goresky-Hingston product on \(\mathbb {C}P^n\) and \(\mathbb {H}P^n\).

The proof of a central Lemma of Sect. 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.

2 Completing manifolds and Morse theory

We start by introducing the notion of a completing manifold following the expositions in [8] and [16].

Let X be a Hilbert manifold and let \(f :\mathbb {X}\rightarrow \mathbb {R}\) 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 \(\textrm{H}_{\bullet }(X^{\le a},X^{<a})\) is well known. It holds that

$$\begin{aligned} \textrm{H}_{\bullet }(X^{\le a},X^{<a}) \cong \textrm{H}_{\bullet -k}(B) \end{aligned}$$

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

([16], Definition 6.1) Let X be a Hilbert manifold and let f be a smooth real-valued 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 \(l = \textrm{dim}(B)\). Assume that k and l are both finite. A completing manifold for B is a closed, orientable manifold \(\Gamma \) of dimension \(k + l\) with an embedding \(\varphi : \Gamma \rightarrow X^{\le a}\) such that the following holds. There is an l-dimensional submanifold L such that \(\varphi |_L\) maps L homeomorphically onto B and there is a retraction map \(p:\Gamma \rightarrow L\). Furthermore, the embedding \(\varphi \) induces a map of pairs

$$\begin{aligned} \varphi :(\Gamma ,\Gamma \setminus L) \rightarrow (X^{\le a}, X^{<a}). \end{aligned}$$

Remark 2.2

  1. (1)

    This definition of a completing manifold is actually the one of a strong completing manifold in [16]. Since all cases that we consider in this article satisfy the assumption of this stronger version we limit our attention to this situation.

  2. (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 Sect. 4.

Recall that if \(f:M\rightarrow N\) is a map between oriented manifolds then the Gysin map

$$\begin{aligned} f_!:\textrm{H}_j(N)\rightarrow \textrm{H}_{j+\textrm{dim}(M)-\textrm{dim}(N)}(M) \end{aligned}$$

is given by

Here \(PD_B\) stands Poincaré duality on the manifold B. The Gysin map \(p_!:\textrm{H}_i(L)\rightarrow \textrm{H}_{i+k}(\Gamma )\) is clearly a right inverse to the Gysin map \(s_!:\textrm{H}_i(\Gamma )\rightarrow \textrm{H}_{i-k}(L)\) where \(s:L\hookrightarrow \Gamma \) is the embedding of L into \(\Gamma \) 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 [5, Theorem VI.11.3]. This shows that the map \(\textrm{H}_{\bullet }(\Gamma )\rightarrow \textrm{H}_{\bullet }(\Gamma ,\Gamma {\setminus } L)\) is surjective. See also [8, Remark 7]. In particular this observation leads to the following result.

Proposition 2.3

( [16], 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

$$\begin{aligned} \textrm{H}_{\bullet }(X^{\le a}) \cong \textrm{H}_{\bullet }(X^{<a}) \oplus \textrm{H}_{\bullet }(X^{\le a},X^{<a}) \cong \textrm{H}_{\bullet }(X^{<a}) \oplus \textrm{H}_{\bullet -k}(B). \end{aligned}$$

If the homology of the sublevel set \(X^{\le a}\) is isomorphic to the direct sum

$$\begin{aligned} \textrm{H}_{\bullet }(X^{\le a}) \cong \textrm{H}_{\bullet }(X^{<a}) \oplus \textrm{H}_{\bullet }(X^{\le a},X^{<a}) \end{aligned}$$
(2.1)

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 \((X^{\le a},X^{<a})\) vanish. If every critical submanifold has a completing manifold, it follows that the function f is perfect. Using completing manifolds Ziller shows in [18] that the energy function on the free loop space of a compact symmetric space is a perfect Morse-Bott function. Note that he uses \(\mathbb {Z}_2\)-coefficients in general, since there are issues with orientability for some spaces. We will describe these completing manifolds for \(M = \mathbb {C}P^n\) and \(M = \mathbb {H}P^n\) in detail in Sect. 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.

3 The string topology coproduct

In this section we introduce the string topology coproduct. We closely follow the definition of the coproduct given in [9]. Let M be an oriented closed N-dimensional Riemannian manifold. We denote the unit interval by \(I = [0,1]\). Let

$$\begin{aligned} PM = \big \{ \gamma : I\rightarrow M\,|\, \gamma \,\,\text {absolutely continuous}, \,\, \int _0^1 |\dot{\gamma }(t)|^2 \,\textrm{d}t < \infty \big \} \end{aligned}$$

be the set of absolutely continuous curves in M such that their derivative is square integrable. See [12, Definition 2.3.1] for the definition of absolutely continuous curves in a manifold. We define the free loop space of M to be

$$\begin{aligned} \Lambda M = \{\gamma \in PM \,|\, \gamma (0 ) = \gamma (1)\} \end{aligned}$$

and this is in fact a submanifold of PM. The manifold M itself can be embedded into \(\Lambda M\) via the trivial loops, see [11, Proposition 1.4.6]. On the path space PM we consider the length functional

$$\begin{aligned} {\mathcal {L}}: PM \rightarrow [0,\infty ),\qquad {\mathcal {L}}(\gamma ) = \sqrt{ \int _0^1 |\dot{\gamma }(t) |^2 \textrm{d}t } \end{aligned}$$
(3.1)

which is a continuous function on PM, see [12, Theorem 2.3.20]. Moreover, it is smooth on \(PM{\setminus } M\). If we restrict \({\mathcal {L}}\) to the free loop space \(\Lambda M\) it turns out that the non-trivial critical points of \({\mathcal {L}}\) 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 \(\epsilon >0\) smaller than the injectivity radius of M. Then the diagonal \(\Delta M\subseteq M\times M\) has a tubular neighborhood given by

$$\begin{aligned} U_M = \{ (p,q)\in M\times M \,|\, \textrm{d}(p,q) < \epsilon \}. \end{aligned}$$

Here, \(\textrm{d}\) is the distance function on M induced by the Riemannian metric. For an \(\epsilon _0> 0\) such that \(\epsilon _0<\epsilon \) we set

$$\begin{aligned} U_{M,\ge \epsilon _0} = \{(p,q)\in U_M\,|\, \textrm{d}(p,q)\ge \epsilon _0\}. \end{aligned}$$

As Hingston and Wahl argue, see [9, Section 1.3], the Thom class in \(\textrm{H}^{N}(TM,TM\setminus M)\) induces a Thom class \(\tau _M\in \textrm{H}^{N}(U_M,U_{M,\ge \epsilon _0})\). On the free loop space we consider the space

$$\begin{aligned} F_{\Lambda } = \{(\gamma ,s)\in \Lambda M\times I\,|\,\gamma (s) = \gamma (0)\}. \end{aligned}$$

We set

$$\begin{aligned} U_{\Lambda }= & {} \{ (\gamma ,s) \in \Lambda M\times I\,|\, \textrm{d}(\gamma (0),\gamma (s))< \epsilon \} \qquad \text {and} \\ U_{\Lambda ,\ge \epsilon _0}= & {} \{ (\gamma ,s)\in U_{\Lambda }\,|\, \textrm{d}(\gamma (0),\gamma (s))\ge \epsilon _0\}. \end{aligned}$$

The set \(U_{\Lambda }\) is an open neighborhood of \(F_{\Lambda }\). Define the evaluation map \(\textrm{ev}_{\Lambda }: \Lambda \times I\rightarrow M\times M\) by

$$\begin{aligned} \textrm{ev}_{\Lambda }(\gamma ,s) = (\gamma (0),\gamma (s)). \end{aligned}$$

This yields a map of pairs

$$\begin{aligned} \textrm{ev}_{\Lambda }:(U_{\Lambda },U_{\Lambda ,\ge \epsilon _0}) \rightarrow (U_M,U_{M,\ge \epsilon _0}). \end{aligned}$$

We define the class

$$\begin{aligned} \tau _{\Lambda } = \textrm{ev}_{\Lambda }^* \tau _M \in \textrm{H}^N(U_{\Lambda },U_{\Lambda ,\ge \epsilon _0}). \end{aligned}$$

Furthermore, there is a retraction map

$$\begin{aligned} R_{GH}:U_{\Lambda }\rightarrow F_{\Lambda }. \end{aligned}$$

We refer to [9, Section 1.5] for its precise definition. Finally, consider the cutting map

$$\begin{aligned} \textrm{cut}:F_{\Lambda }\rightarrow \Lambda M\times \Lambda M \end{aligned}$$

which maps a point \((\gamma ,s)\in \Lambda \times I\) with \(\gamma (0)=\gamma (s)\) to the pair of loops \( (\gamma |_{[0,s]},\gamma |_{[s,1]}) \) and reparametrizes both loops such that they are again defined on the unit interval I. Note that the cutting map actually factors through maps

where \(\Lambda M\times _M \Lambda M\) is the figure-eight space

$$\begin{aligned} \Lambda M\times _M \Lambda M = \{ (\gamma ,\sigma )\in \Lambda M\times \Lambda M\,|\, \gamma (0) = \sigma (0) \}. \end{aligned}$$

With this preparation we can now define the string topology coproduct. Let [I] be the positively oriented generator of \(\textrm{H}_1(I,\partial I)\) with respect to the standard orientation of the unit interval. In order to shorten notation we shall also write \(\Lambda \) for the free loop space \(\Lambda M\).

Definition 3.1

The string topology coproduct is defined as the map

Remark 3.2

Let M be a closed oriented manifold.

  1. (1)

    Note that the cap product with the class \(\tau _{\Lambda }\) is a particular relative cap product. This relative cap product is defined in Appendix B where we also study some basic properties. See also [9, Appendix A].

  2. (2)

    Hingston and Wahl define an algebraic loop coproduct, see [9, 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 \(\mathbb {F}\) be a field and assume that the homology of \(\Lambda M\) is of finite type. Let \(\alpha \in \textrm{H}^i(\Lambda ,M;\mathbb {F})\) and \(\beta \in \textrm{H}^j(\Lambda ,M;\mathbb {F})\) be relative cohomology classes, then the Goresky-Hingston product is defined to be the unique cohomology class in \(\textrm{H}^{i+j+N-1}(\Lambda ,M;\mathbb {F})\) such that

Remark 3.4

Let M be a closed oriented manifold.

  1. (1)

    The Goresky-Hingston product can also be defined intrinsically, see [7]. However, in this article we shall study properties of this product only via the duality with the string topology coproduct.

  2. (2)

    As for the string topology coproduct, Hingston and Wahl define a sign-corrected version of the Goresky-Hingston product in [9]. 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 \( M = \mathbb {C}P^n\) or \(M = \mathbb {H}P^n\) at this point.

  • In Sect. 4 we will study the critical manifolds \(\Sigma _k\), \(k\in \mathbb {N}\) in \(\Lambda M\) of the length functional \({\mathcal {L}}:\Lambda M\rightarrow \mathbb {R}\).

  • In Sect. 5 we construct the completing manifolds \(\Gamma _k\).

  • We shall see that the manifold \(\Gamma _k\), \(k\in \mathbb {N}\) can also serve as a completing manifold for critical submanifolds in the figure-eight space \(\Lambda M\times _M\Lambda M\).

  • We determine the cohomology ring of \(\Gamma _k\) in Sect. 6. We can then explicitly compute the Gysin map and obtain a set of generators for the homology \(\textrm{H}_{\bullet }(\Lambda M,M)\).

  • In Sect. 7 we will then replicate all the steps in the definition of the coproduct on the manifold \(\Gamma _k\).

  • We will pull back the class \(\tau _{\Lambda }\) to a class which can be described in terms of the cohomology of \(\Gamma _k\) and compute the cap product with this class.

  • Then one sees that under the cutting map \(\widetilde{\textrm{cut}}:F_{\Lambda }\rightarrow \Lambda M \times _M\Lambda M\) we get homology classes which we can identify with classes coming from the manifold \(\Gamma _k\) seen as a completing manifold in the figure-eight space \(\Lambda M\times _M \Lambda M\).

4 Critical manifolds in the free loop space of projective spaces

In this section we describe the completing manifolds on the loop space of \(\mathbb {C} P^n\) and \(\mathbb {H}P^n\) in detail. We will first study the critical manifolds of \(\Lambda M\) and \(\Lambda M\times _M \Lambda M\) with respect to the length functional \({\mathcal {L}}\). Then we define the completing manifolds \(\Gamma _k\), \(k\in \mathbb {N}\) and show that \(\Gamma _k\) can serve as a completing manifold both in \(\Lambda M\) as well as in \(\Lambda M\times _M \Lambda M\). Finally, we describe the cohomology ring of \(\Gamma _k\) in detail and give a set of explicit generators of \(\textrm{H}_{\bullet }(\Lambda M,M)\).

From now on let \(\mathbb {K}= \mathbb {C}\) or \(\mathbb {K}= \mathbb {H}\). In case \(\mathbb {K}= \mathbb {C}\) we set

$$\begin{aligned} N = 2n, \quad \text {and}\quad \lambda = 1 \end{aligned}$$

and in case \(\mathbb {K}= \mathbb {H}\) we set

$$\begin{aligned} N = 4n, \quad \text {and}\quad \lambda = 3. \end{aligned}$$

We consider the free loop space of \(M = \mathbb {K}P^n\) where we consider M as a symmetric space. The symmetric Riemannian metric on M induces a length functional \({\mathcal {L}}:\Lambda M\rightarrow \mathbb {R}\), see Eq. (3.1) and it is well-known that this is a Morse-Bott function on \(\Lambda M\). Moreover, the index and the nullity of all critical manifolds are finite, see [18]. 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 \(p_0\in M\). 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 \(k\in \mathbb {N}\) 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 [18] that the group G acts transitively on \(\Sigma ^a\) and this action equals the canonical action of G on the unit tangent bundle. Let \(\gamma \in \Sigma ^a\) be a closed geodesic at level \(a = kl\) with \(\gamma (0) = p_0\). Then there is an underlying prime geodesic \(\sigma \in \Sigma ^l\) such that \(\gamma = \sigma ^k\). Denote the isotropy group of \(\gamma \) with respect to the action of G on \(\Sigma ^a\) by \(K_{\gamma }\). In particular, this is a closed subgroup of K. Then we have

$$\begin{aligned} SM \cong \Sigma ^a \cong G/K_{\gamma }. \end{aligned}$$

Furthermore, there is an induced action of the group K on \(\Sigma ^a\) and the orbit of \(\gamma \) is

$$\begin{aligned} K.\gamma \cong K/K_{\gamma } \cong \mathbb {S}^{N-1}, \end{aligned}$$
(4.1)

which is the fiber of SM over \(p_0\). In the following we will always write \(\Sigma ^k\) for the critical submanifold at level kl instead of \(\Sigma ^{kl}\). The index of \(\Sigma ^k\) is

$$\begin{aligned} \textrm{ind}(\Sigma ^k) = k\lambda + (k-1)(N-1) \end{aligned}$$

see [7, p. 167] and the nullity is equal to the dimension of \(\Sigma ^k\), i.e.

$$\begin{aligned} \textrm{null}(\Sigma ^k) = 2N-1. \end{aligned}$$

On the figure-eight space \(\Lambda M\times _M\Lambda M\) we consider the length function

$$\begin{aligned} {\mathcal {L}}_2:\Lambda M\times _M \Lambda M\rightarrow [0,\infty ),\quad {\mathcal {L}}_2(\eta _1,\eta _2) = {\mathcal {L}}(\eta _1) + {\mathcal {L}}(\eta _2) \end{aligned}$$

for \((\eta _1,\eta _2)\in \Lambda M\times _M\Lambda M\). The critical manifolds in \(\Lambda M\times _M \Lambda M\) are the sets of the form

$$\begin{aligned} \Sigma _m\times _M \Sigma _{k-m} = \Sigma _m\times \Sigma _{k-m} \cap \Lambda M\times _M \Lambda M, \quad k,m\in \mathbb {N}_0, \,\, k\ge m \end{aligned}$$

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.

$$\begin{aligned} \Sigma _m\times _M \Sigma _{k-m} = \{ (\gamma _1,\gamma _2)\in \Sigma _m\times \Sigma _{k-m}\,|\, \gamma _1(0) = \gamma _2(0)\}. \end{aligned}$$

At level kl the critical set is

$$\begin{aligned} M\times _M \Sigma _k \,\, \sqcup \,\, \Sigma _1 \times _M \Sigma _{k-1} \,\, \sqcup \,\, \ldots \,\, \sqcup \,\, \Sigma _{k-1}\times _M \Sigma _1\,\, \sqcup \,\, \Sigma _k\times _M M. \end{aligned}$$

Note that by the relative construction of the coproduct the components \(M\times _M \Sigma _k\) and \(\Sigma _k\times _M M\) will not show up in the course of the proof so we will not deal with them. See also Remark 7.6.

Lemma 4.1

The length function \({\mathcal {L}}_2:\Lambda M\times _M \Lambda M\rightarrow [0,\infty )\) satisfies the Palais-Smale condition and is a Morse-Bott function. Moreover, we have

$$\begin{aligned} \textrm{ind}_{\Lambda \times _M \Lambda }(\eta _1,\eta _2) = \textrm{ind}_{\Lambda }(\eta _1) + \textrm{ind}_{\Lambda }(\eta _2) \end{aligned}$$

and

$$\begin{aligned} \textrm{null}_{\Lambda \times _M\Lambda }(\eta _1,\eta _2) = \textrm{null}_{\Lambda }(\eta _1) + \textrm{null}(\eta _2) - N \end{aligned}$$

for a critical point \((\eta _1,\eta _2)\in \Lambda M\times _M \Lambda M\) of the function \({\mathcal {L}}_2\).

Proof

The function

$$\begin{aligned} {\mathcal {L}}':\Lambda \times \Lambda \rightarrow [0,\infty ), \quad (\gamma _1,\gamma _2)\mapsto {\mathcal {L}}(\gamma _1)+{\mathcal {L}}(\gamma _2) \end{aligned}$$

clearly satisfies the Palais-Smale condition. Since \(\Lambda M \times _M \Lambda M\) is a closed submanifold of \(\Lambda \times \Lambda \) and since \({\mathcal {L}}_2\) is the restriction of \(\widetilde{{\mathcal {L}}}\) it therefore follows that \({\mathcal {L}}_2\) also satisfies the Palais-Smale condition.

In order to show that \({\mathcal {L}}_2\) is a Morse-Bott function, we need to show the following property. Let \((\eta _1,\eta _2)\in \Lambda M\times _M \Lambda M\) be a critical point of \({\mathcal {L}}_2\) and assume that it belong to the critical submanifold of the form \(\Sigma ^a\times _M \Sigma ^b\) where \(\Sigma ^a\) and \(\Sigma ^b\) are critical submanifolds in \(\Lambda M\) with respect to the Morse-Bott function \({\mathcal {L}}\). We need to show that the null space \(T_{(\eta _1,\eta _2)}^0 \Lambda M\times _M \Lambda M\) is equal to the tangent space \(T_{(\eta _1,\eta _2)}\Sigma ^a \times _M \Sigma ^b\subseteq T_{\eta _1,\eta _2} \Lambda M\times _M \Lambda M\). It is clear that we have

$$\begin{aligned} T_{(\eta _1,\eta _2)}\Sigma ^a \times _M \Sigma ^b\subseteq T_{\eta _1,\eta _2}^0 \Lambda M\times _M \Lambda M. \end{aligned}$$

Arguing as in [12, Section 2.5] one can see that \(T_{(\eta _1,\eta _2)}^0\Lambda M\times _M \Lambda M\) can be characterized as

$$\begin{aligned} T_{(\eta _1,\eta _2)}^0\Lambda \times _M \Lambda{} & {} = \{ (\xi _1,\xi _2)\in T_{\eta _1} \Lambda \oplus T_{\eta _2}\Lambda \,|\, \xi _1 (0) \\{} & {} = \xi _2(0), \,\,\xi _1,\,\xi _2\,\,\text {periodic Jacobi fields} \}. \end{aligned}$$

Ziller shows in [18, Section 2] that all periodic Jacobi fields along closed geodesics in a compact symmetric space are restrictions of Killing vector fields. Let \(\xi _1,\xi _2\) be periodic Jacobi fields along \(\eta _1\) and \(\eta _2\), respectively. We assume without loss of generality that \(\eta _1(0) =\eta _2(0) = p_0\) is the basepoint. If \(\xi _1(0) = \xi _2(0) = 0\) 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 \(K\times K\) on \(M\times M\) clearly preserves the diagonal \(\Delta M\) it is clear that \((\xi _1,\xi _2)\in T_{(\eta _1,\eta _2)}\Sigma ^a\times _M \Sigma ^b\). If we have \(\xi _1(0) = \xi _2(0) \ne 0\) then let us assume that \(\nabla \xi _1(0) = \nabla \xi _2(0) = 0\). In this case one sees as in [18, page 8] that both Jacobi fields are restrictions of the same Killing field, since the Killing field are determined by the element \(\xi _1(0)\) in this case. Hence these Jacobi fields can be understood as restrictions of a Killing field of the diagonal group action \(G\times M\times M\rightarrow M\times M\). Therefore in this case we also see that \((\xi _1,\xi _2)\in T_{(\eta _1,\eta _2)}\Sigma ^a\times _M \Sigma ^b\). Since the Jacobi fields of the above two types form a basis of \( T_{(\eta _1,\eta _2)}^0\Lambda \times _M \Lambda \) this shows the inclusion

$$\begin{aligned} T_{(\eta _1,\eta _2)}^0\Lambda \times _M \Lambda \subseteq T_{(\eta _1,\eta _2)}\Sigma ^a \times _M \Sigma ^b. \end{aligned}$$

Consequently, \({\mathcal {L}}_2\) 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

$$\begin{aligned} \textrm{ind}(\eta _1) + \textrm{ind}(\eta _2)= & {} \textrm{ind}_{\Omega \times \Omega }((\eta _1,\eta _2)) \\\le & {} \textrm{ind}_{\Lambda \times _M \Lambda }((\eta _1,\eta _2)) \\\le & {} \textrm{ind}_{\Lambda \times \Lambda }((\eta _1,\eta _2)) \\\le & {} \textrm{ind}(\eta _1) + \textrm{ind}(\eta _2) \end{aligned}$$

and thus we see that the inequalities are all equalities. This completes the proof. \(\square \)

For \(i,j\ge 1\) such that \(i+j = k\) we have

$$\begin{aligned} \Sigma _i\times _M \Sigma _j \cong SM\times _M SM = \{ (u,v)\in SM\times SM\,|\, \textrm{pr}(u) = \mathrm {pr(v)} \} \end{aligned}$$

where \(\textrm{pr}:SM\rightarrow M\) is the canonical projection of the unit sphere bundle of M. Moreover, the projection onto the first factor makes \(SM\times _M SM\) into a sphere bundle over SM which admits a global section

$$\begin{aligned} SM\rightarrow SM\times _M SM, \quad u\mapsto (u,u),\quad u\in SM. \end{aligned}$$

Since later on we shall use the cohomology ring of \(SM\times _M SM\), we prove the following Lemma.

Lemma 4.2

The rational cohomology ring of \(SM\times _M SM\) is isomorphic to

$$\begin{aligned} \textrm{H}^{\bullet }(SM\times _M SM) \cong \frac{\mathbb {Q}[{\alpha },{\beta },{\xi }]}{({\alpha }^n,{\beta }^2,{\xi }^2)} \end{aligned}$$

where \(\textrm{deg}({\alpha }) = \lambda + 1\), \(\textrm{deg}({\beta }) = N+\lambda \) and \(\textrm{deg}({\xi }) = N-1\).

Proof

From the Gysin sequence for \(SM\rightarrow M\) we know that

$$\begin{aligned} \textrm{H}^{\bullet }(SM) \cong \frac{\mathbb {Q}[\alpha ,\beta ]}{(\alpha ^{n},\beta ^2)} \end{aligned}$$

with \(\textrm{deg}(\alpha ) = \lambda +1\) and \(\textrm{deg}(\beta ) = N+\lambda \). The manifold \(SM\times _M SM\) is the total space of a sphere bundle over SM with a global section, therefore it follows from the corresponding Gysin sequence that

$$\begin{aligned} \textrm{H}^{\bullet }(SM\times _M SM) \cong \textrm{H}^{\bullet }(SM)\otimes \Lambda [{\xi }] \end{aligned}$$

with a generator \({\xi }\) of degree \(\textrm{deg}({\xi }) = N-1\). This proves the claim. \(\square \)

For \(i\in \{0,\ldots ,n-1\}\) we denote the homology class dual to the class \({\alpha }^i\) by \([{a}_i]\) and the dual of \({\alpha }^i{\beta }\) by \([{a}_i{b}]\).

Before we turn to the completing manifolds, let us note a property of the conjugate points along the closed geodesics in M. With \(\gamma \) and \(\sigma \) as before, note that there is precisely one conjugate point \(\sigma (\tfrac{1}{2}) = a\) along \(\sigma \), see [2, Proposition 3.35]. Moreover, the index of \(\sigma \) is equal to \(\lambda \), 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 [2, Proposition 3.35] and [18]. Denote the isotropy group of this point with respect to the action of K by \(K_a\). It is well-known that

$$\begin{aligned} \textrm{dim}(K_a) > \textrm{dim}(K_{\gamma }), \end{aligned}$$

see [18, Theorem 4], and that \(\textrm{dim}(K_a) - \textrm{dim}(K_{\gamma })\) is equal to the index of \(\sigma \) both as a geodesic loop in \(\Omega M\) as well as a closed geodesic in \(\Lambda M\).

Lemma 4.3

The homogeneous space \(K_a/K_{\gamma }\) is diffeomorphic to the sphere \(\mathbb {S}^{\lambda }\).

Proof

By [2, 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

$$\begin{aligned} \textrm{Cut}(\mathbb {K}P^n) \cong \mathbb {K}P^{n-1}, \end{aligned}$$

see again [2, Proposition 3.35]. The set of tangent cut points

$$\begin{aligned} S = \{v\in T_{p_0} M\,|\, \textrm{exp}_{p_0}(v) \,\,\text {is the cut point of} \,\, t \mapsto \exp _p(tv) \} \end{aligned}$$

is well-known to be the round sphere \(\mathbb {S}^{N-1}\). Moreover, the exponential map induces a fiber bundle \(\exp _{p_0}:\mathbb {S}^{N-1}\rightarrow \mathbb {K}P^{n-1}\), see [2, Proposition 3.37]. These are just the well-known fibrations of spheres over projective space, so it follows that the fiber is \(\mathbb {S}^{\lambda }\). See also [2, Theorem 5.29] for details. We can understand these objects as homogeneous spaces, i.e. as we know from Eq. (4.1) we have \(K/K_{\gamma }\cong \mathbb {S}^{N-1}\) and it is clear that \(K/K_a\) is diffeomorphic to the cut locus \(\mathbb {K}P^{n-1}\). Therefore we see that that the fiber \(K_{a}/K_{\gamma }\) of the fiber bundle \(K/K_{\gamma }\rightarrow K/K_a\) is diffeomorphic to \(\mathbb {S}^{\lambda }\). \(\square \)

5 Construction of Ziller’s completing manifolds

In this section we describe Ziller’s completing manifolds, see [18]. We describe the manifolds and the respective embeddings in detail.

Fix a closed geodesic \(\gamma = \sigma ^k\in \Lambda M\) of multiplicity k starting at the basepoint \(p_0\in M\). Here, \(\sigma \) is the underlying prime closed geodesic. Consider the product

$$\begin{aligned} W_k = G\times K_a \times K\times K_a \times K \ldots \times K_a \end{aligned}$$

with 2k factors in total. Throughout this section, we will follow the convention that the first element in the tuple

$$\begin{aligned} (g_0,x_1,\ldots ,x_{2k-1})\in W_k \end{aligned}$$

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 \(K_{\gamma }\) on \(W_k\) given by

$$\begin{aligned} \chi : W_k \times K_{\gamma }^{2k}\rightarrow & {} W_k \\ ( (g_{0},x_{1},\ldots ,x_{2k-1}), (h_0,\ldots ,h_{2k-1}))\mapsto & {} (g_0 h_0,h_0^{-1} x_1 h_1,\ldots , h_{2k-2}^{-1} x_{2k-1} h_{2k-1}). \end{aligned}$$

This action is free and proper and we consider the quotient space

$$\begin{aligned} \Gamma _k = W_k/(K_{\gamma }^{2k}). \end{aligned}$$

There is an embedding \(f_k :\Gamma _k \rightarrow \Lambda M\) given by

$$\begin{aligned} {f_{k}}([g_0,x_1,\ldots ,x_{2k-1}])(t) = \left\{ \begin{array}{ll} g_{0}. \gamma (t), &{} 0\le t\le \tfrac{1}{2k} \\ g_0x_1.\gamma (t), &{} \tfrac{1}{2k}\le t \le \tfrac{2}{2k} \\ \,\,\,\,\, \vdots &{} \,\,\,\,\, \vdots \\ g_0x_1\ldots x_{2k-1}.\gamma (t), &{} \tfrac{2k-1}{2k}\le t \le 1 \end{array}\right. . \end{aligned}$$

Note that the critical submanifold \(\Sigma _k \cong G/K_{\gamma }\) can be seen as a submanifold of \(\Gamma _k\) via the embedding

$$\begin{aligned} s_{L,k}:G/K_{\gamma } \rightarrow \Gamma _k,\qquad s_{L,k}([g]) = [ g,e,\ldots ,e ] \,\,\,\text {for}\,\,\,[g]\in G/K_{\gamma }. \end{aligned}$$

We define \(L_k = s_{L,k}(G/K_{\gamma })\) and will identify \(L_k\) and \(G/K_{\gamma }\) in the following possibly without making the identification explicit. There is a submersion

$$\begin{aligned}{} & {} p_{L,k} :\Gamma _k \rightarrow G/K_{\gamma }, \,\,\,\,p_{L,k}([g_0,x_1,\ldots ,x_{2k-1}]) = [g_0] \nonumber \\{} & {} \quad \text {for}\,\,\, [g_0,x_1,\ldots ,x_{2k-1}] \in \Gamma _k \end{aligned}$$
(5.1)

and it is clear that \( p_{L,k}\circ s_{L,k} = {{\,\textrm{id}\,}}_{G/K_{\gamma }} \). Moreover, we see that the composition \(f_k\circ s_{L,k}\) is given by

$$\begin{aligned} f_k \circ s_{L,k} ([g])(t) = g.\gamma (t) \qquad \text {for}\,\,\,g\in G, \,\, t\in [0,1]. \end{aligned}$$

Hence, the map \(f_k \circ s_{L,k}\) is precisely the diffeomorphism \(G/K_{\gamma }\cong \Sigma ^k\). Note that the only closed geodesics in the image of \(f_k\) are precisely the closed geodesics in the critical submanifold \(\Sigma _k\). All other loops in \(\textrm{im}(f_k)\) 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 \({\mathcal {L}}\) for all \(\gamma \in \textrm{im}(f_k)\) which are not in \(\Sigma _k\). Consequently, if we compose the embedding \(f_k\) with an arbitrary short gradient flow of the length functional, we obtain a map of pairs

$$\begin{aligned} (\Gamma _k, \Gamma _k \setminus L_k) \rightarrow (\Lambda M^{\le kl},\Lambda M^{<kl}). \end{aligned}$$

Since

$$\begin{aligned} \textrm{dim}(\Gamma _k)= & {} N + k(N-1) + k (\textrm{dim}(K_a)-\textrm{dim}(K_{\gamma })) \\= & {} \textrm{ind}(\gamma ) + 2N-1 = \textrm{ind}(\gamma ) + \textrm{dim}(\Sigma _k) \end{aligned}$$

we have shown that \(\Gamma _k\) is a completing manifold for \(\Sigma _k\) if we prove that it is orientable. We will see the orientability later.

Remarkably, \(\Gamma _k\) can also serve as a completing manifold in the Hilbert manifold \( \Lambda M\times _M \Lambda M\). With k and \(\gamma \) as above, fix \(1\le m\le k-1\) and consider the critical point

$$\begin{aligned} (\gamma _1,\gamma _2) = (\sigma ^m,\sigma ^{k-m})\in \Lambda M\times _M \Lambda M, \end{aligned}$$

where \(\sigma \) is the underlying prime geodesic of \(\gamma \). The component of the critical set at level k in \(\Lambda M\times _M \Lambda M\) that contains \((\gamma _1,\gamma _2)\) is \(\Sigma _m \times _M \Sigma _{k-m}\). Note that

$$\begin{aligned} \textrm{ind}((\gamma _1,\gamma _2)){} & {} = \textrm{ind}(\gamma _1) + \textrm{ind}(\gamma _2) = k \, \textrm{ind}(\sigma ) + (k-2)(N-1) \\{} & {} = \textrm{ind}(\gamma ) - (N-1), \end{aligned}$$

see Lemma 4.1. We want to see how \(SM\times _M SM\) can be embedded into \(\Gamma _k\). Consider the right-action of \(K_{\gamma }\times K_{\gamma }\) on \(G\times K\)

$$\begin{aligned} \chi ' :(G\times K) \times (K_{\gamma }\times K_{\gamma }) \rightarrow G\times K \end{aligned}$$

given by

$$\begin{aligned} \chi '((g_0,x_{2m}),(h_0,h_{2m})) = (g_0 h_0,h_0^{-1} x_{2m} h_{2m}) \end{aligned}$$

for \(g_0\in G,x_{2\,m}\in K\) and \(h_0,h_{2\,m}\in K_{\gamma }\). Like the action \(\chi \) above, this is a free and proper right action and we consider the quotient space \( {\mathcal {V}} = G\times K/ \chi ' \).

Lemma 5.1

The manifold \({\mathcal {V}}\) is diffeomorphic to \(SM\times _M SM\).

Proof

Recall that there is a diffeomorphism \( G/K_{\gamma } \rightarrow SM\) induced by the transitive action of G on SM. In particular we see that

$$\begin{aligned} SM\times _M SM \cong \{ ([g_1],[g_2])\in G/K_{\gamma }\times G/K_{\gamma }\,|\, g_1^{-1} g_2\in K\}. \end{aligned}$$
(5.2)

Moreover, let \(E\subseteq G\times G\) be the submanifold

$$\begin{aligned} E = \{ (g_1,g_2)\in G\times G\,|\, g_1^{-1} g_2\in K\}. \end{aligned}$$

It is clear that there is a submersion \(E\rightarrow SM\times _M SM\) given by \((g_1,g_2) \mapsto ([g_1],[g_2])\). Now, define maps

$$\begin{aligned} {\widetilde{\varphi }}:G\times K\rightarrow E \quad \text {and}\quad {\widetilde{\psi }}:E\rightarrow G\times K \end{aligned}$$

by setting

$$\begin{aligned} {\widetilde{\varphi }}(g,k) = (g,gk) \quad \text {for}\,\,\, g\in G,k\in K \end{aligned}$$

and

$$\begin{aligned} {\widetilde{\psi }}(g_1,g_2) = (g_1,g_1^{-1}g_2) \quad \text {for} \,\,\, (g_1,g_2)\in E. \end{aligned}$$

These maps factor through the submersions \(G\times K\rightarrow {\mathcal {V}}\) and \(E\rightarrow SM\times _M SM\) and therefore induce smooth maps

$$\begin{aligned} \varphi :{\mathcal {V}}\rightarrow SM\times _M SM \quad \text {and}\quad \psi :SM\times _M SM\rightarrow {\mathcal {V}}. \end{aligned}$$

It is a direct computation that they are inverses of each other. \(\square \)

Observe that there is an embedding

$$\begin{aligned} s_{V,m} :{\mathcal {V}} \hookrightarrow \Gamma _k \end{aligned}$$

given by

$$\begin{aligned} s_{V,m}( [g_0,x_{2m}] ) = [g_0,e,\ldots ,e,x_{2m},e,\ldots ,e] \in \Gamma _k \end{aligned}$$

where \(x_{2m}\) appears at the 2m’th position. We denote the image of \({\mathcal {V}}\) under this embedding by V. There is a submersion \(p_{V,m}:\Gamma _k \rightarrow {\mathcal {V}}\) given by

$$\begin{aligned} p_{V,m} ([g_0,x_1,\ldots ,x_{2k-1}]) = [g_0,x_1x_2\ldots x_{2m}] \in {\mathcal {V}}. \end{aligned}$$

It is clear that \( p_{V,m}\circ s_{V,m} = {{\,\textrm{id}\,}}_{{\mathcal {V}}} \). We define a map \(F_{k,m}:\Gamma _k \rightarrow \Lambda M\times _M \Lambda M\) as follows. Let

$$\begin{aligned} F_{k,m}([g_0,x_1,\ldots ,x_{2k-1}]) = (\eta _1,\eta _2) \quad \text {for}\quad [g_0,x_1,\ldots ,x_{2k-1}]\in \Gamma _k \end{aligned}$$
(5.3)

where

$$\begin{aligned} \eta _1(t) = \left\{ \begin{array}{ll} g_0.\gamma _1(t ), &{} 0\le t\le \tfrac{1}{2m} \\ \vdots \\ g_0 x_1\ldots x_{2m-1}.\gamma _1(t ), &{} \tfrac{2m-1}{2m}\le t\le 1 \end{array}\right. \end{aligned}$$

and

$$\begin{aligned} \eta _2(t) = \left\{ \begin{array}{ll} g_0 x_1\ldots x_{2m-1}x_{2m}.\gamma _2(t), &{} 0\le t\le \tfrac{1}{2k-2m} \\ \vdots \\ g_0 x_1\ldots x_{2k-1}.\gamma _2(t),&{} \tfrac{2k-2m-1}{2k-2m}\le t\le 1. \end{array}\right. \end{aligned}$$

It can be checked directly that \(F_{k,m}\) is a continuous embedding.

Lemma 5.2

The embedding \(F_{k,m} :\Gamma _k\rightarrow \Lambda M\times _M \Lambda M\) maps V homeomorphically onto the critical set \(\Sigma _m \times _M \Sigma _{k-m}\). Moreover, the set of critical points in \(\Lambda M\times _M \Lambda M\) in the image of \(F_{k,m}\) is precisely the set \(\Sigma _m\times _M \Sigma _{k-m}\).

Proof

We show that the diagram

commutes, where the map \(i_m\) is the inclusion of \(SM\times _M SM\) into \(\Lambda M\times _M \Lambda M\) as the critical set \(\Sigma _m\times _M \Sigma _{k-m}\). If we identify \(SM\times _M SM\) with the subspace of \(G/K_{\gamma }\times G/K_{\gamma }\) as in Eq. (5.2) then

$$\begin{aligned} i_m([g_1],[g_2]) = (g_1.\gamma _1,g_2.\gamma _2)\qquad \text {for} \,\,\,\, [g_1],[g_2]\in G/K_{\gamma }\,\,\,\text {with}\,\, g_1g_2^{-1}\in K. \end{aligned}$$

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 \(F_{k,m}\). \(\square \)

If we compose \(F_{k,m}\) with an arbitrarily short flow of the gradient flow of the length functional \({\mathcal {L}}_2\) on \(\Lambda M\times _M \Lambda M\) we obtain a map of pairs

$$\begin{aligned} (\Gamma _k,\Gamma _k\setminus V) \rightarrow ((\Lambda M\times _M \Lambda M)^{\le kl},(\Lambda M\times _M \Lambda M)^{<kl}). \end{aligned}$$

Furthermore, we have

$$\begin{aligned} \textrm{dim}(\Gamma _k) = \textrm{ind}(\gamma _1,\gamma _2) + \textrm{dim}(\Sigma _m\times _M \Sigma _{k-m}) \end{aligned}$$

so if we show that \(\Gamma _k\) is orientable we see that \(\Gamma _k\) is a completing manifold for \( \Sigma _m\times _M \Sigma _{k-m}\).

Note that in \(\Lambda M\times _M \Lambda M\) we can also use the fiber product \(\Gamma _m\times _M \Gamma _{k-m}\) as a completing manifold where the fiber product is taken with respect to the evaluation map

$$\begin{aligned} \textrm{ev}_k :\Gamma _k \rightarrow M,\quad [g_0,x_1,\ldots ,x_{2k-1}] \mapsto g_0 K\in G/K\cong M \end{aligned}$$

for \(1\le l \le k\). Then one takes the map

$$\begin{aligned} (f_m,f_{k-m}):\Gamma _m\times _M \Gamma _{k-m}\rightarrow \Lambda M\times _M \Lambda M \end{aligned}$$

as an embedding of \(\Gamma _m\times _M \Gamma _{k-m}\) and can check that this is again a completing manifold. To conclude this section we want to show that the completing manifolds \(\Gamma _k\) and \(\Gamma _m\times _M \Gamma _{k-m}\) are equivalent. Define a map \(\Phi _m:\Gamma _k\rightarrow \Gamma _m\times _M \Gamma _{k-m}\) by

$$\begin{aligned}{} & {} \Phi _m ([g_0,x_1,\ldots , x_{2k-1}] ) \\{} & {} \quad = ( [g_0,x_1,\ldots ,x_{2m-1}],[g_0 x_1\ldots x_{2m-1} x_{2m}, x_{2m+1},\ldots , x_{2k-1}]) \end{aligned}$$

for \([g_0,x_1,\ldots ,x_{2k-1}]\in \Gamma _k\). Note that this is a well-defined and smooth map since it descends from an equivariant map \(W_k\rightarrow W_m\times W_{k-m}\). Similarly, we define \(\Psi _m:\Gamma _m\times _M \Gamma _{k-m}\rightarrow \Gamma _k\) by

$$\begin{aligned}{} & {} \Psi ([g_0,x_1,\ldots ,x_{2m-1}],[g_{2m},\ldots ,x_{2k-1}]) \\ {}{} & {} = [g_0,x_1,\ldots ,x_{2m-1}, (g_0x_1\ldots x_{2m-1})^{-1}g_{2m},x_{2m+1},\ldots , x_{2k-1}] \end{aligned}$$

for \([g_0,x_1,\ldots ,x_{2\,m-1}]\in \Gamma _m, [g_{2\,m},\ldots ,x_{2k-1}]\in \Gamma _{k-m}\) with \(g_0^{-1}g_{2\,m}\in K\). One checks again that \(\Psi _m\) is well-defined and smooth.

Lemma 5.3

The completing manifolds \(\Gamma _k\) and \(\Gamma _m\times _M \Gamma _{k-m}\) for the critical submanifold \(\Sigma _m\times _M \Sigma _{k-m}\) are equivalent in the sense that the diagrams

and

commute. Here, \(p_{L,m}:\Gamma _m\rightarrow SM\) and \(p_{L,k-m}:\Gamma _{k-m}\rightarrow SM\) are the submersions which are used for the completing manifold structure in the free loop space, see Eq. (5.1). In particular \(\Phi \) is a diffeomorphism.

Proof

This can be checked directly by unwinding the definitions. \(\square \)

6 Cohomology of the completing manifolds

The manifold \(\Gamma _k\) is closely related to the K-cycles in the sense of Bott and Samelson, see [6]. In this section we shall describe its homology and cohomology following the discussions of the cohomology of the K-cycles by Bott and Samelson [6] and by Araki [1]. Recall that \(\Gamma _k\) is defined as the quotient of \(W = (G\times K_a)\times (K\times K_a)^{k-1}\) modulo the action of \(K_{\gamma }^{2k}\) via

$$\begin{aligned} \chi ((g_0,x_1,\ldots ,x_{2k-1}), (h_0,\ldots ,h_{2k-1})) = (g_0h_0,h_0^{-1}x_1 h_1^{-1},\ldots , h_{2k-2}^{-1}x_{2k-1} h_{2k-1}). \end{aligned}$$

If we have any product of subgroups \(K_i\subseteq G\), \(i\in \{1,\ldots ,m\}\) such that \(K_{\gamma }\subseteq K_i\) for all \(i\in \{1,\ldots ,m\}\) there is an action of \((K_{\gamma })^m\) on this product given by

$$\begin{aligned} ((x_1,\ldots ,x_m), (h_1,\ldots ,h_m)) \mapsto (x_1 h_1, h_1^{-1}x_2 h_2,\ldots , h_{m-1}^{-1} x_m h_m) \end{aligned}$$

for \((x_1,\ldots ,x_m)\in K_1\times \ldots K_m\) and \((h_1,\ldots ,h_m)\in (K_{\gamma })^m\). We denote the quotient by

$$\begin{aligned} K_1 \times _{K_{\gamma }} K_2 \times _{K_{\gamma }} \ldots \times _{K_{\gamma }} (K_m/K_{\gamma }). \end{aligned}$$

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 \(0\le i_1< i_2< \ldots < i_m \le 2k-1\) be integers and set \(K_{0} = G\), \(K_j = K\) for j even, \(j\ge 2\) and \(K_j = K_a\) for j odd. Then the manifold

$$\begin{aligned} \Gamma ^{i_1 i_2\ldots i_m} = K_{i_1}\times _{K_{\gamma }} K_{i_2}\times _{K_{\gamma }}\ldots \times _{K_{\gamma }} (K_{i_m}/K_{\gamma }) \end{aligned}$$

can be embedded into \(\Gamma _k\) via a map \(s_{i_1\ldots i_m}:\Gamma ^{i_1\ldots i_m}\hookrightarrow \Gamma _k\).

Proof

We define a map \(\sigma _{i_1\ldots i_m}:K_{i_1}\times K_{i_2}\times \ldots K_{i_m}\rightarrow W_k\) by

$$\begin{aligned} \sigma _{i_1\ldots i_m}(x_{i_1},\ldots x_{i_m}) = (e,\ldots ,e,x_{i_1},e,\ldots , e,x_{i_2},e,\ldots , e,x_{i_m},e\ldots ,e) \end{aligned}$$

where \(x_{i_j}\) is at position \(i_j\) for each \(j\in \{1,\ldots ,m\}\). If

$$\begin{aligned} (x_{i_1},\ldots ,x_{i_m})\in K_{i_1}\times \ldots \times K_{i_m} \quad \text {and}\quad (k_1,\ldots ,k_m)\in K_{\gamma }^m \end{aligned}$$

we have

$$\begin{aligned}{} & {} \sigma _{i_1\ldots i_m} ((x_{i_1},\ldots ,x_{i_m}). (k_1,\ldots ,k_m)) \\ {}{} & {} \quad = (e,\ldots , e, x_{i_1} k_1, e,\ldots , e, k_1^{-1} x_{i_2} k_2,\ldots ) \\ {}{} & {} \quad = (e,\ldots , e, x_{i_1} k_1, k_1^{-1} k_1,\ldots , k_1^{-1} k_1, k_1^{-1} x_{i_2} k_2,k_2^{-1}k_2,\ldots ) \\{} & {} \quad = \chi ( \sigma _{i_1\ldots i_m}(x_{i_1},\ldots ,x_{i_m}), (e,\ldots ,e,k_1,k_1,\ldots ,k_1,k_2,\ldots ,k_2,\ldots )). \end{aligned}$$

Hence, \(\sigma _{i_1\ldots i_m}\) is equivariant with respect to the action of \(K_{\gamma }^m\) on \(K_{i_1}\times \ldots \times K_{i_m}\) and the action \(\chi \) of \(K_{\gamma }^{2k}\) on \(W_k\). Therefore this yields a smooth map

$$\begin{aligned} s_{i_1\ldots i_m}:\Gamma ^{i_1\ldots i_m}\rightarrow \Gamma _k. \end{aligned}$$
(6.1)

It is easy to check that this is an embedding. \(\square \)

Let \(P = (i_1,i_2,\ldots ,i_m)\) with non-negative integers \(0\le i_1< i_2< \ldots < i_m \le 2k-1\). Then we say that \(\Gamma ^P = \Gamma ^{i_1\ldots i_m}\) is a sub-K-cycle of \(\Gamma _k\). From now on we will always identify a sub-K-cycle \(\Gamma ^P\) with its image in \(\Gamma _k\) under the embeddings constructed in Lemma 6.1. The manifold \({\mathcal {V}} \cong V\) which we defined above is an example of a sub-K-cycle. If \(P = (0,1,\ldots ,m)\) for some \(m\le 2k-2\), then there are submersions

$$\begin{aligned} p_{\Gamma ,m} :\Gamma _k\rightarrow \Gamma ^P \end{aligned}$$

given by

$$\begin{aligned} p_{\Gamma ,m}([g_0,x_1,\ldots ,x_{2k-1}]) = [g_0,x_1,\ldots ,x_m] \quad \text {for}\,\,\, [g_0,x_1,\ldots ,x_{2k-1}]\in \Gamma _k. \end{aligned}$$

Hence, we get a chain of submersions

$$\begin{aligned} \Gamma _k \rightarrow \Gamma ^{0\ldots 2k-2} \rightarrow \ldots \rightarrow \Gamma ^{01} \rightarrow \Gamma ^0 = G/K_{\gamma }. \end{aligned}$$

As shown in [1, Theorem 2.4] all these submersions are fiber bundles. Moreover, each fiber bundle has a section which is given by the map

$$\begin{aligned} \Gamma ^{0\ldots m-1}\hookrightarrow \Gamma ^{0\ldots m}, \quad [g_0,x_1,\ldots ,x_{m-1}] \mapsto [g_0,x_1,\ldots ,x_{m-1},e]. \end{aligned}$$

Definition 6.2

Let

be a sequence of manifolds with each \(\pi _i:E_i\rightarrow E_{i+1}\) being a sphere bundle. Then we say that \(E_0\) is an iterated sphere bundle over B.

Proposition 6.3

The K-cycle \(\Gamma _k\) is an iterated sphere bundle via the maps

$$\begin{aligned} \Gamma _k \rightarrow \Gamma ^{0\ldots 2k-2} \rightarrow \ldots \rightarrow \Gamma ^{01} \rightarrow \Gamma ^0 = G/K_{\gamma }. \end{aligned}$$

Proof

The fibers of the iterated fiber bundle are either \(K/K_{\gamma }\) of \(K_a/K_{\gamma }\). We know that \(K/K_{\gamma } \cong \mathbb {S}^{N-1}\) and \(K_a/K_{\gamma } \cong \mathbb {S}^{\lambda }\), see Lemma 4.3. Hence, \(\Gamma _k\) is an iterated sphere bundle over \(SM\cong G/K_{\gamma }\). \(\square \)

In the situation of an iterated sphere bundle \(E_0\rightarrow E_1\rightarrow \ldots \rightarrow E_m = B\), 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 \(\Gamma _k\). Note that throughout the article we have fixed an orientation on M. In particular this induces an orientation on the \(\epsilon \)-sphere around the basepoint \(p_0\). We shall denote the generator of its fundamental class by \([\mathbb {S}^{N-1}_{\epsilon }]\). Furthermore, if \(t\in I\), define \(\textrm{ev}_t:\Lambda M \rightarrow M\) to be the map \(\textrm{ev}_t(\gamma ) = \gamma (t)\) for \(\gamma \in \Lambda M\).

Proposition 6.4

The cohomology ring of \(\Gamma _k\) is isomorphic to

$$\begin{aligned} \textrm{H}^{\bullet }(\Gamma _k) \cong \frac{\mathbb {Q}[\alpha ,\beta ,\xi _1,\ldots ,\xi _{2k-1}]}{ (\alpha ^{n},\beta ^2,\xi _1^2,\ldots ,\xi _{2k-1}^2) } \end{aligned}$$

where \(\textrm{deg}(\alpha ) = \lambda +1\), \(\textrm{deg}(\beta ) = N+\lambda \), \(\textrm{deg}(\xi _{2i+1}) = \lambda \) for \(i=0,\ldots ,k-1\) and \(\textrm{deg}(\xi _{2i}) = N-1\) for \(i=1,\ldots ,k-1\). In particular \(\Gamma _k\) is orientable.

Furthermore, the class \(\xi _{2i}\) and the dual class \([x_{2i}]\) in homology for \(i\in \{1,\ldots ,k-1\}\) can be chosen such that the following holds. For \(i\in \{1,\ldots ,k-1\}\) one can choose a fundamental class \([\mathbb {S}^{N-1}]\) of \(\mathbb {S}^{N-1}\) such that \([x_{2i}] = (s_{2i})_*[\mathbb {S}^{N-1}]\) and such that

$$\begin{aligned} (\textrm{ev}_{t_i}|_{f_k(\Gamma _k)} \circ f_k \circ s_{2i})_*[\mathbb {S}^{N-1}] = [\mathbb {S}^{N-1}_{\epsilon }] \end{aligned}$$

where \(\textrm{ev}_{t_i}|_{f_k(\Gamma _k)}\) is understood as a map \(\textrm{ev}_{t_i}|_{f_k(\Gamma _k)}:f_k(\Gamma _k) \rightarrow B_{p_0}{\setminus }\{p_0\}\) for \(t_i\in (\tfrac{i}{k}, \tfrac{i}{k} + \delta ) \) for some small \(\delta > 0\) and where \(s_{2i}:\mathbb {S}^{N-1}\rightarrow \Gamma _k\) is the embedding constructed in Lemma 6.1.

Proof

As we have seen before it follows from the Gysin sequence of \(SM\rightarrow M\) that

$$\begin{aligned} \textrm{H}^{\bullet }(SM) \cong \frac{\mathbb {Q}[\alpha ,\beta ]}{(\alpha ^{n},\beta ^2)} \end{aligned}$$

with \(\textrm{deg}(\alpha ) = \lambda +1\) and \(\textrm{deg}(\beta ) = N+\lambda \). One can now determine the cohomology ring of \(\Gamma _k\) by induction along the steps of the iterated sphere bundle. Note that for a k-sphere bundle \(E\rightarrow B\) with k odd and which admits a global section one has

$$\begin{aligned} \textrm{H}^{\bullet }(E) \cong \textrm{H}^{\bullet }(B) \otimes \Lambda _{\mathbb {Q}}[\xi ] \end{aligned}$$

where \(\textrm{deg}(\xi ) = k\). 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 \(\xi _i\). As Araki argues using Gysin sequences, see [1, Section 2], the homology class dual to the class \(\xi _l\) for \(l\in \{1,\ldots ,2k-1\}\) can be described as follows. Let \(s_l:K_l/K_{\gamma }\hookrightarrow \Gamma _k\) be the embedding as in Lemma 6.1. Then if we consider an orientation class \([K_l/K_{\gamma }]\in \textrm{H}_{\bullet }(K/K_{\gamma })\) we have

$$\begin{aligned} (s_l)_*[K_l/K_{\gamma }] = \pm [x_{l}]. \end{aligned}$$

If \(l=2i\) we have \(K_{2i}/K_{\gamma }\cong \mathbb {S}^{N-1}\). Let \(t_i\in (\tfrac{i}{k}\tfrac{i}{k}+\delta )\) with \(\delta >0\) small then it can be seen directly that the map

$$\begin{aligned} \textrm{ev}_{t_i} \circ f_k\circ s_{2i}:\mathbb {S}^{N-1} \rightarrow M \end{aligned}$$

maps \(\mathbb {S}^{N-1}\) homeomorphically onto \(\mathbb {S}^{N-1}_{\epsilon '}\) for some small \(\epsilon '>0\). 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 \([\mathbb {S}^{N-1}]\) and correspondingly the class \(\xi _{2i}\) and its dual \([x_{2i}]\) such that

$$\begin{aligned}{}[x_{2i}] = (s_{2i})_*[\mathbb {S}^{N-1}]\quad \text {and}\quad (\textrm{ev}_{t_i}\circ f_k\circ s_{2i})_*[\mathbb {S}^{N-1}] = [\mathbb {S}^{N-1}_{\epsilon '}] \end{aligned}$$

where we consider \(\textrm{ev}_{t_i}\circ f_k\circ s_{2i}\) as a map \(\mathbb {S}^{N-1} \rightarrow \mathbb {S}^{N-1}_{\epsilon '}\). \(\square \)

In the previous proposition we saw that the manifolds \(\Gamma _k\), \(k\in \mathbb {N}\) are orientable. This completes the proof that the \(\Gamma _k\) are in fact completing manifolds. We sum this up in the next corollary.

Corollary 6.5

Let \(M = \mathbb {K}P^n\) be a complex or quaternionic projective space and let \(k,m\in \mathbb {N}\) with \(k> m\).

  1. (1)

    The manifold \(\Gamma _k\) with the embedding \(f_k\) is a completing manifold for the critical set \(\Sigma _k\) in \(\Lambda M\).

  2. (2)

    The manifold \(\Gamma _k\) with the embedding \(F_{k,m}\) is a completing manifold for the critical set \(\Sigma _m\times _M \Sigma _{k-m}\) in \(\Lambda M\times _M \Lambda M\).

Note that the Corollary implies the perfectness of the Morse-Bott function \({\mathcal {L}}\) on \(\Lambda M\). This is one of the main results in [18].

We now make the following notation convention for the generators in homology. As shown above the classes

$$\begin{aligned} \alpha ^i \beta ^j \xi _1^{l_1} \ldots \xi _{2k-1}^{l_{2k-1}}, \quad i\in \{0,\ldots ,n-1\},\,\, j,l_i\in \{0,1\} \,\, \text {for}\,\,i\in \{1,\ldots ,2k-1\} \end{aligned}$$

generate the cohomology of \(\Gamma _k\) additively. If \(l_{i_1},\ldots , l_{i_p} = 1\) and \(l_i = 0\) otherwise then we denote the dual of \( \alpha ^i \beta ^j \xi _1^{l_1} \ldots \xi _{2k-1}^{l_{2k-1}}\) in homology by

$$\begin{aligned}{}[a_i x_{i_1\ldots i_p}] \in \textrm{H}_{\bullet }(\Gamma _k) \quad \text {if}\quad j=0 \quad \text {and} \quad [a_i b x_{i_1\ldots i_p}]\quad \text {if}\quad j = 1. \end{aligned}$$

Recall that at level kl the level homology \(\textrm{H}_{\bullet }(\Lambda M^{\le kl},\Lambda M^{<kl})\) is isomorphic to the homology of the critical submanifold \(L_k \cong SM\). Moreover, as we have seen in Sect. 2 the map

$$\begin{aligned} (p_{L,k})_! :\textrm{H}_{i-\lambda _k}(L_k) \rightarrow \textrm{H}_i(\Gamma _k) \end{aligned}$$

is injective, where

$$\begin{aligned} \lambda _k = \textrm{ind}(\gamma ) = k\lambda + (k-1)(N-1). \end{aligned}$$

Hence, we obtain the generators of \(\textrm{H}_{\bullet }(\Lambda M)\) which come from level k by considering the map \((p_{L,k})_!\). Recall that \(L_k \cong SM\) and as we have seen its cohomology ring is

$$\begin{aligned} \textrm{H}^{\bullet }(L_k;\mathbb {Q}) \cong \frac{\mathbb {Q}[{\alpha },{\beta }]}{({\alpha }^n,{\beta }^2)} \end{aligned}$$

with \(\textrm{deg}({\alpha }) = \lambda + 1\) and \(\textrm{deg}({\beta }) = N + \lambda \). In homology we choose dual generators and denote them by \([{a}_0],\ldots , [{a}_{n-1}]\in \textrm{H}_{\bullet }(SM;\mathbb {Q})\) with \(\textrm{deg}([{a_i}]) = (\lambda +1) i\) and \([{a}_0{b}],\ldots ,[{a}_{n-1}{b}] \in \textrm{H}_{\bullet }(SM;\mathbb {Q})\) with \(\textrm{deg}([{a_i}{b}]) = N +\lambda + i(\lambda + 1)\). In particular, we can choose these generators such that under the embedding

$$\begin{aligned} s_{L,k}:G/K_{\gamma } \cong SM \hookrightarrow \Gamma _k \end{aligned}$$

we have

$$\begin{aligned} (s_{L,k})_* [{a}_i] = [ a_i] \quad \text {and}\quad (s_{L,k})_*[{a}_i{b}] = [a_i b] \end{aligned}$$

and in cohomology

$$\begin{aligned} (p_{L,k})^*{\alpha }^i = \alpha ^i \quad \text {and} \quad (p_{L,k})^* {\beta } = \beta \end{aligned}$$
(6.2)

where \(p_{L,k}:\Gamma _k\rightarrow L_k\) is the retraction. The above formulas also justify the misuse of notation, since e.g. the cohomology class \(\alpha \) has a double meaning, but as \((p_{L,k})_*:\textrm{H}^{\bullet }(L_k)\rightarrow \textrm{H}^{\bullet }(\Gamma _k)\) is injective, it is reasonable to identify \(\alpha \in \textrm{H}^{\bullet }(L_k)\) with its image under this injection. We choose the orientation of \(L_k\) by choosing the class \([{a}_{n-1}{b}]\in \textrm{H}_{2N-1}(L_k)\) as fundamental class for all \(k\in \mathbb {N}\) and as fundamental class for \(\Gamma _k\) we choose the class

$$\begin{aligned}{}[a_{n-1}b x_{1\ldots 2k-1}]\in \textrm{H}_{2N-1 + \lambda _k}(\Gamma _k). \end{aligned}$$

Recall that the Gysin map \((p_{L,k})_!:\textrm{H}_{\bullet }(L_k)\rightarrow \textrm{H}_{\bullet +\lambda }(\Gamma _k)\) is defined as the composition

Using Proposition 6.4 we can now compute the map \((p_{L,k})_!\).

Lemma 6.6

With the above notation the following equations hold

$$\begin{aligned} (p_{L,k})_!([{a}_i]) = - [a_ix_{1\ldots 2k-1}] \in \textrm{H}_{\lambda _k + i(\lambda +1)}(\Gamma _k) \end{aligned}$$

and

$$\begin{aligned} (p_{L,k})_!([{a}_i{b}]) = [a_i b x_{1\ldots 2k-1}] \in \textrm{H}_{\lambda _k + (i+1)(\lambda +1) + N-1 }(\Gamma _k). \end{aligned}$$

Proof

We just consider the first case, the second case is analogous. The Poincaré dual of \([{a}_i]\in \textrm{H}_{\bullet }(L_k)\) is the cohomology class \({\alpha }^{n-1-i}{\beta }\in \textrm{H}^{\bullet }(L_k)\). By Eq. (6.2) we have

$$\begin{aligned} (p_{L,k})^* ({\alpha }^{n-1-i}{\beta }) = \alpha ^{n-1-i}\beta \in \textrm{H}^{\bullet }(\Gamma _k). \end{aligned}$$

We now need to compute the Poincaré dual \(X = PD_{\Gamma _k}(\alpha ^{n-1-i}\beta )\) of this latter class. We compute the Kronecker pairing

$$\begin{aligned} \langle \alpha ^i \xi _1 \ldots \xi _{2k-1}, X \rangle= & {} \langle \alpha ^i\xi _1\ldots \xi _{2k-1} ,\, \alpha ^{n-1-i}\beta \cap [\Gamma _k] \rangle \\= & {} \langle \alpha ^i \xi _1\ldots \xi _{2k-1} \, \cup \, \alpha ^{n-1-i}\beta ,\, [\Gamma _k]\rangle . \end{aligned}$$

Now, since \(\alpha \) is of even degree and \(\beta \) is of odd degree we get by graded commutativity

$$\begin{aligned} \alpha ^i \xi _1\ldots \xi _{2k-1} \, \cup \, \alpha ^{n-1-i}\beta = - \alpha ^{n-1}\beta \xi _1\ldots \xi _{2k-1} \end{aligned}$$

and therefore

$$\begin{aligned} \langle \alpha ^i \xi _1 \ldots \xi _{2k-1}, X \rangle = -1. \end{aligned}$$

It follows that \(X = - [a_i b x_{1\ldots 2k-1}]\). \(\square \)

We define classes

$$\begin{aligned} A_k^i = (f_k)_* \big ( - [a_ix_{1\ldots 2k-1}] \big ) \in \textrm{H}_{\lambda _k + i(\lambda +1)}(\Lambda M) \end{aligned}$$

and

$$\begin{aligned} B_k^i = (f_k)_* \big ( [a_i b x_{1\ldots 2k-1}] \big ) \in \textrm{H}_{\lambda _k + (i+1)(\lambda +1) + N-1 }(\Lambda M) \end{aligned}$$

for \(k\in \mathbb {N}\) and \(i\in \{0,\ldots ,n-1\}\). Note that the degree of all \(A^i_k\) is odd, while the degree of all \(B_k^i\) 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

$$\begin{aligned}{} & {} \{ A_k^i \in \textrm{H}_{\bullet }(\Lambda M) \,|\,k\in \mathbb {N}, i\in \{0,\ldots ,n-1\} \} \,\,\cup \,\, \\{} & {} \{ B_k^i\in \textrm{H}_{\bullet }(\Lambda M) \,|\,k\in \mathbb {N}, i\in \{0,\ldots ,n-1\} \} \end{aligned}$$

in the relative homology \(\textrm{H}_{\bullet }(\Lambda M,M)\).

7 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 \(\textrm{H}_{\bullet }(\Lambda M, M)\) via the completing manifolds \(\Gamma _k\), \(k\in \mathbb {N}\). We therefore express all the steps in the definition of the coproduct in intrinsic terms of the manifolds \(\Gamma _k\).

First, we want to pull back the class \(\tau _{\Lambda }\in \textrm{H}^N(U_{\Lambda },U_{\Lambda ,\ge \epsilon _0})\) via \(f_k\) to a class which can be described in terms of the cohomology of \(\Gamma _k\). We consider the preimage \((f_k,{{\,\textrm{id}\,}}_I)^{-1}(U_{\Lambda })\subseteq \Gamma _k\times I\). This set can be described explicitly as follows. Since all loops in the image \(f_k(\Gamma _k)\) are broken geodesics, there is a small \(\delta >0\) such that

$$\begin{aligned}{} & {} (f_k,{{\,\textrm{id}\,}}_I)^{-1}(U_{\Lambda }) \\{} & {} \quad = \Gamma _k \times \Big ( [0,\delta ) \cup (\tfrac{2}{2k}-\delta ,\tfrac{2}{2k}+\delta ) \cup \ldots \cup (\tfrac{2k-2}{2k}-\delta ,\tfrac{2k-2}{2k}+\delta )\cup (1-\delta ,1]\Big ). \end{aligned}$$

This is because every other conjugate point on a closed geodesic starting at the basepoint is the basepoint itself. Clearly, \(\delta >0\) is so small that the open intervals are disjoint. Similarly, there is a \(\delta _0>0\) with \(\delta _0<\delta \) such that

$$\begin{aligned} (f_k,{{\,\textrm{id}\,}}_I)^{-1}(U_{\Lambda ,\ge \epsilon _0}) = \Gamma _k \times \Big ( [\delta _0,\delta ) \cup (\tfrac{2}{2k}-\delta , \tfrac{2}{2k}-\delta _0]\cup [\tfrac{2}{2k}+\delta _0,\tfrac{2}{2k} +\delta ) \cup \\ \ldots \cup (\tfrac{2k-2}{2k}-\delta ,\tfrac{2k-2}{2k}-\delta _0]\cup [ \tfrac{2k-2}{2k}+\delta _0,\tfrac{2k-2}{2k}+\delta ) \cup (1-\delta ,1-\delta _0] \Big ). \end{aligned}$$

To make the bookkeeping easier, let us define

$$\begin{aligned} I_m = (\tfrac{2m}{2k}-\delta ,\tfrac{2m}{2k}+\delta ) \end{aligned}$$

and

$$\begin{aligned} J_m = (\tfrac{2m}{2k} - \delta ,\tfrac{2m}{2k}-\delta _0] \cup [\tfrac{2m}{2k}+\delta _0,\tfrac{2m}{2k}+\delta ) \end{aligned}$$

for \(m = 1,\ldots ,k-1\). We set

$$\begin{aligned} U_{\Gamma _k} = (f_k,{{\,\textrm{id}\,}}_I)^{-1}(U_{\Lambda })\quad \text { and} \quad U_{\Gamma _k,\ge \epsilon _0} = (f_k,{{\,\textrm{id}\,}}_I)^{-1}(U_{\Lambda ,\ge \epsilon _0}), \end{aligned}$$

then we have

$$\begin{aligned}{} & {} (U_{\Gamma _k},U_{\Gamma _k,\ge \epsilon _0}) \\{} & {} \quad = \Gamma _k \times \Big ( ([0,\delta ),[\delta _0,\delta )) \sqcup \bigsqcup _{m=1}^{k-1} (I_m,J_m) \sqcup ((1-\delta ,1],(1-\delta ,1-\delta _0] ) \Big ). \end{aligned}$$

The pairs

$$\begin{aligned} ([0,\delta ),[\delta _0,\delta )) \quad \text {and}\quad ((1-\delta ,1],(1-\delta ,1-\delta _0]) \end{aligned}$$

have trivial homology for obvious reasons. For \(m\in \{1,\ldots ,k-1\}\) we have

$$\begin{aligned} H_i(I_m,J_m) \cong \left\{ \begin{array}{ll} \mathbb {Q}, &{} i=1\\ 0&{} \text {else}. \end{array}\right. \end{aligned}$$

We choose positively oriented generators \([I_m]\) of \(\textrm{H}_1(I_m,J_m)\) and dual cohomology classes

$$\begin{aligned} \eta _m\in \textrm{H}^1(I_m, J_m). \end{aligned}$$

We now want express the cohomology class \(\tau _k = (f_k,{{\,\textrm{id}\,}}_I)^{*}\tau _{\Lambda } \in \textrm{H}^{N}(U_{\Gamma _k},U_{\Gamma _k,\ge \epsilon _0})\) in terms of the cohomology of \(\Gamma _k\). Recall that the cap product which we use in the definition of the string topology coproduct is a particular relative version of the ordinary cap product. We refer to Appendix B for details, see also [9, Appendix A].

Lemma 7.1

The pullback of the class \(\tau _{\Lambda }\in \textrm{H}^{N}(U_{\Lambda },U_{\Lambda ,\ge \epsilon _0})\) under the map

$$\begin{aligned} (f_k,{{\,\textrm{id}\,}}_I):(U_{\Gamma _k},U_{\Gamma _k,\ge \epsilon _0}) \rightarrow (U_{\Lambda },U_{\Lambda ,\ge \epsilon _0}) \end{aligned}$$

is given by

$$\begin{aligned} \tau _k = (f_k,{{\,\textrm{id}\,}}_I)^*\tau _\Lambda = \sum _{m=1}^{k-1} \xi _{2m}\times \eta _m. \end{aligned}$$

This key lemma is proved in Appendix A. In the following let Y be one of the classes

$$\begin{aligned} A_k^i = (f_k)_* \big ( - [a_ix_{1\ldots 2k-1}] \big ) \in \textrm{H}_{\bullet }(\Lambda M) \end{aligned}$$

or

$$\begin{aligned} B_k^i = (f_k)_* \big ( [a_i b x_{1\ldots 2k-1}] \big ) \in \textrm{H}_{\bullet }(\Lambda M) \end{aligned}$$

for \(k\in \mathbb {N}\), \(i\in \{0,\ldots ,n-1\}\). We write

$$\begin{aligned} Y = (f_k)_* X \end{aligned}$$

where \(X\in \textrm{H}_{\bullet }(\Gamma _k)\) is the respective homology class in \(\Gamma _k\). In order to compute \(\vee Y\) we need to consider

$$\begin{aligned} \tau _{\Lambda } \cap (Y\times [I])= & {} (f_k,{{\,\textrm{id}\,}}_I)_*( (f_k,{{\,\textrm{id}\,}}_I)^*\tau _{\Lambda } \cap (X\times [I])) \\ {}= & {} (f_k,{{\,\textrm{id}\,}}_I)_*(\tau _k\cap (X\times [I]) ) \end{aligned}$$

by naturality of the cap product, see Proposition B.1. As seen in Lemma 7.1, we obtain

$$\begin{aligned} \tau _k \cap (X\times [I]) = \sum _{m=1}^{k-1} \big ( (\xi _{2m}\times \eta _m) \cap (X\times [I]) \big ). \end{aligned}$$

By the compatibility of the cross and the cap product, see Proposition B.2, we have

$$\begin{aligned} (\xi _{2m}\times \eta _m)\cap (X\times [I]) = (-1)^{\textrm{deg}(X)} (\xi _{2m}\cap X)\times (\eta _m\cap [I]). \end{aligned}$$

By the construction of the relative cap product we see that

$$\begin{aligned} \eta _m\cap [I] = [t_m]\in \textrm{H}_0(I_m) \end{aligned}$$

where \([t_m]\) is a generator of \(\textrm{H}_0(I_m)\), see Example B.3.

Lemma 7.2

The relative cap product yields

$$\begin{aligned} (\xi _{2m}\times \eta _m)\cap \big ( - [a_i x_{1\ldots 2k-1}] \times [I] \big ) = - [a_i x_{1\ldots 2m-1\, 2m+1\ldots 2k-1}] \times [t_m] \end{aligned}$$

and

$$\begin{aligned} (\xi _{2m}\times \eta _m)\cap \big ( [a_i b x_{1\ldots 2k-1}] ) \times [I] \big ) = - [a_i b x_{1\ldots 2m-1\, 2m+1\ldots 2k-1}] \times [t_m] \end{aligned}$$

Proof

Using Proposition 6.4 we have

$$\begin{aligned} \xi _{2m} \cap \big ( - [a_i x_{1\ldots 2k-1}] \big ) = [a_i x_{1\ldots 2m-1 \, 2m+1\ldots 2k-1}] \end{aligned}$$

and

$$\begin{aligned} \xi _{2m} \cap [a_i b x_{1\ldots 2k-1}] = - [a_i b x_{1\ldots 2m-1 \, 2m+1\ldots 2k-1}] \end{aligned}$$

and this yields the claim. \(\square \)

For convenience of notation we shall write

$$\begin{aligned} (\xi _{2m}\times \eta _m)\cap (X\times [I]) = X_m\times [t_m] \end{aligned}$$

and plug in the respective classes later using the above Lemma. Then we have

$$\begin{aligned} \tau _k\cap (X\times [I]) = \sum _{m=1}^{k-1} X_m\times [ t_m]. \end{aligned}$$

Fix an \(m\in \{1,\ldots ,k-1\}\). To finish the computation of \(\vee Y\), we need to determine the effect of the retraction map \(\textrm{R}_{GH}\) and of the cutting map on \(X_m\times [t_m] \). First note that the diagram

commutes, where \(\sigma _m:I_m\rightarrow \{\tfrac{2m}{2k}\}\) is the constant map. To complete the computation, we need to characterize

$$\begin{aligned} \textrm{cut}_*(f_k,{{\,\textrm{id}\,}}_I)_* (X_m\times [\tfrac{2m}{2k}]) \in \textrm{H}_{\bullet }(\Lambda M \times \Lambda M,\Lambda M\times M\cup M\times \Lambda M). \end{aligned}$$

A direct computation shows that the map

$$\begin{aligned} \widetilde{\textrm{cut}} \circ f_k \, :\,\, \Gamma _k\times \{\tfrac{2m}{2k}\} \rightarrow \Lambda M\times _M \Lambda M \end{aligned}$$

is equal to the map \(F_{k,m}:\Gamma _k \rightarrow \Lambda M\times _M \Lambda M\), which was defined in Eq. (5.3), up to the obvious identification \(\Gamma _k\cong \Gamma _k\times \{\tfrac{2m}{2k}\}\). This shows that

$$\begin{aligned} \mathrm {\widetilde{cut}}_*(f_k,{{\,\textrm{id}\,}}_I)_* ( X_m\times [\tfrac{2m}{2k}]) = (F_{k,m})_* X_m. \end{aligned}$$

Here and in the following \([\tfrac{2m}{2k}]\in \textrm{H}_0(\{\tfrac{2m}{2k}\})\) denotes the canonical generator. We now want to express the class \((\iota \circ F_{k,m})_* X_m \in \textrm{H}_{\bullet }(\Lambda M\times \Lambda M)\) as a product of the generators \(A_k^i\) and \(B_k^i\). Here, \(\iota :\Lambda M\times _M \Lambda M\hookrightarrow \Lambda M\times \Lambda M\) 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

where \(\textrm{Th}:\textrm{H}_i(\Gamma _m\times \Gamma _{k-m})\) is the map

Here, U is a tubular neighborhood of \(SM\times SM\) in \(\Gamma _m\times \Gamma _{k-m}\) and

$$\begin{aligned} \textrm{Thom}:\textrm{H}_{\bullet }(U,U\setminus SM\times SM) \rightarrow \textrm{H}_{\bullet -\lambda _k +(N-1)}(SM\times SM) \end{aligned}$$

is the Thom isomorphism. The map \(\textrm{Th}'\) is defined analogously. In particular, we note that the normal bundle of \(SM\times _M SM \hookrightarrow \Gamma _m\times _M \Gamma _{k-m}\) is the pullback of the normal bundle of \(SM\times SM\hookrightarrow \Gamma _m\times \Gamma _{k-m}\) along the inclusion \(SM \times _M SM \hookrightarrow SM\times SM\). Therefore the above diagram commutes. Now, note that the map \(\textrm{Th}\) agrees with the Gysin map \((s_m,s_{k-m})_!\). This follows from [5, 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 \((-1)^{c (\lambda _k - (N-1))}\) for some integer \(c\in \mathbb {Z}\). Recall that the index \(\lambda _k\) is odd for all \(k\in \mathbb {N}\). Consequently, the codimension

$$\begin{aligned} \textrm{codim}(SM\times SM\hookrightarrow \Gamma _m\times \Gamma _{k-m}) = \lambda _k - (N-1) \end{aligned}$$

is even for all \(k\in \mathbb {N}\), so we see that the sign \((-1)^{c (\lambda _k - (N-1))}\) is even and thus the maps \(\textrm{Th}\) and \((s_{L,m},s_{L,k-m})_!\) agree. Now, let \(Z\in \textrm{H}_{\bullet }(SM\times _M SM)\). Then we have

$$\begin{aligned} Z = ((p_{L,m},p_{L,k-m}) \, \circ \, (s_{L,m},s_{L,k-m}) )_! \,Z = (s_{L,m},s_{L,k-m})_! \circ (p_{L,m},p_{L,k-m})_! \, Z. \end{aligned}$$

Moreover, let us denote the inclusion \(SM\times _M SM\hookrightarrow SM\times SM\) by \(i_1\) and denote the inclusion \(\Gamma _m\times _M \Gamma _{k-m}\hookrightarrow \Gamma _m\times \Gamma _{k-m}\) by \(i_2\). Then we get

$$\begin{aligned} (p_{L,m},p_{L,k-m})_! \circ (i_1)_* Z= & {} (p_{L,m},p_{L,k-m})_! \circ (i_1)_* (s_{L,m},s_{L,k-m})_! \circ (p_{L,m},p_{L,k-m})_! Z \\ {}= & {} (p_{L,m},p_{L,k-m})_! \circ (s_{L,m},s_{L,k-m})_! \circ (i_2)_* \circ (p_{L,m},p_{L,k-m})_! Z \\ {}= & {} (i_2)_* \circ (p_{L,m},p_{L,k-m})_! Z \end{aligned}$$

and this shows the commutativity of the lower left square. \(\square \)

Recall that \(\Gamma _k\) together with the embedding \(F_{k,m}\) is a completing manifold for the critical set \(\Sigma _m\times _M \Sigma _{k-m}\cong V\). In Lemma 4.2 we determined the cohomology ring of V. We make the following choice of orientation. The generator \(\xi \in \textrm{H}^{N-1}(V)\) is chosen in such a way that it pulls back to the generator \(\xi _{2m}\) under the map \(p_{V,m}\). Note that by the choice of orientations for the classes \(\xi _{2i}\), \(i\in \{1,\ldots ,k-1\}\) in Proposition 6.4 this is a consistent choice. We choose the orientation for V such that \(\alpha ^{n-1}\beta \xi \) is a fundamental cohomology class.

Lemma 7.4

Using the notation for the generators of \(\textrm{H}_{\bullet }(SM\times _M SM)\) as in the paragraph following Lemma 4.2 we have

$$\begin{aligned} (p_{V,m})_! [ {a}_i] = - [a_ix_{1\ldots 2m-1\, 2m+1 \ldots 2k-1}] \end{aligned}$$

and

$$\begin{aligned} (p_{V,m})_![{a}_i{b}] = - [a_i b x_{1\ldots 2m-1 \, 2m+1 \ldots 2k-1}]. \end{aligned}$$

Proof

We only consider the first case, the second one is analogous. The Poincaré dual of \([{a}_i]\) is the class \(\alpha ^{n-1-i}\beta \xi \in \textrm{H}_{\bullet }(V)\). Moreover, by our orientation convention, the pullback of this cohomology class is

$$\begin{aligned} (p_{V,m})^* (\alpha ^{n-1-i}\beta \xi ) = \alpha ^{n-1-i}\beta \xi _{2m}. \end{aligned}$$

Now we need to compute the Poincaré dual of this class in \(\Gamma _{k}\), i.e.

$$\begin{aligned} PD_{\Gamma _k}( \alpha ^{n-1-i}\beta \xi _{2m}) = \alpha ^{n-1-i}\beta \xi _{2m} \cap [\Gamma _k]. \end{aligned}$$

By using the graded commutativity of the cup product, we see that

$$\begin{aligned}{} & {} \langle \alpha ^i \xi _1\ldots \xi _{2m-1}\xi _{2m+1}\ldots \xi _{2k-1}, \alpha ^{n-1-1}\beta \xi _{2m} \cap [\Gamma _k]\rangle \\{} & {} \quad = \langle -\alpha ^{n-1}\beta \xi _1\ldots \xi _{2k-1},[\Gamma _k]\rangle = -1. \end{aligned}$$

Therefore we see that

$$\begin{aligned} \alpha ^{n-1-1}\beta \xi _{2m} \cap [\Gamma _k] = - [a_i x_{1\ldots 2m-1 \, 2m+1\ldots 2k-1}]. \end{aligned}$$

\(\square \)

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 \(A_k^i\) and \(B_k^i\). 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 \(\textrm{H}_{\bullet }(SM)\) under the embedding \(SM\hookrightarrow V\) given by

$$\begin{aligned}{}[g] \mapsto [g,e] \in V,\quad \text {where}\,\,\, [g]\in G/K_{\gamma } \cong SM, \end{aligned}$$

see also Lemma 4.2. The composition

is just the diagonal map \(d:SM\rightarrow SM\times SM\). Hence, we obtain

$$\begin{aligned} \omega ([{a}_i]) = d_* [{a_i}] \quad \text {and}\quad \omega ([{a}_i {b}]) = d_*[{a_i}{ b}] \end{aligned}$$

where \(d:SM\rightarrow SM\times SM\) 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

$$\begin{aligned} d_*[{a_i}] = \sum _{j=0}^i [{a_j}]\times [{a}_{i-j}] \quad \text {and}\quad d_*[{a_i}{ b}] = \sum _{j=0}^i [{a_j}]\times [{a}_{i-j}{b}] + [{a_j} {b}]\times [{a}_{i-j} ]. \end{aligned}$$

We obtain the following final result.

Theorem 7.5

Let \(\mathbb {K}\) be \(\mathbb {C}\) or \(\mathbb {H}\) and consider \(M = \mathbb {K}P^n\). The string topology coproduct on M behaves as follows. We have

$$\begin{aligned} \vee A_k^i = \sum _{m=1}^{k-1} \sum _{j=0}^i A_m^j\times A_{k-m}^{i-j} \end{aligned}$$

and

$$\begin{aligned} \vee B_k^i = \sum _{m=1}^{k-1} \sum _{j=0}^i ( B_m^j \times A_{k-m}^{i-j} -A_m^j \times B_{k-m}^{i-j} ). \end{aligned}$$

Proof

If \(Y = A_{k}^i\), we have

$$\begin{aligned} Y = (f_k)_* ( - [a_ix_{1\ldots 2k-1}]). \end{aligned}$$

By Lemmas 7.1 and 7.2, we have

$$\begin{aligned} \tau _{\Lambda }\cap ( Y\times [I]) = \sum _{m=1}^{k-1} (f_k)_*( -[a_i x_{1\ldots 2m-1 \,2m+1\ldots 2k-1}]) \times [\tfrac{2m}{2k}]. \end{aligned}$$

Then by Lemmas 7.3 and 7.4 we see that

$$\begin{aligned}{} & {} \textrm{cut}_* \big ( \sum _{m=1}^{k-1} (f_k)_*( -[a_i x_{1\ldots 2m-1 \,2m+1\ldots 2k-1}]) \times [\tfrac{2m}{2k}] \big ) \\{} & {} = \sum _{m=1}^{k-1} (f_m,f_{k-m})_*\big ( (p_{L,m},p_{L,k-m})_! d_*[{a_i}] \big ). \end{aligned}$$

Hence we are left with computing the Gysin map of the retraction \((p_{L,m},p_{L,k-m})\). By [5, Proposition VI.14.3] we see that

$$\begin{aligned}{} & {} (p_{L,m},p_{L,k-m})_! (x\times y) = (-1)^{(\textrm{dim}(SM) + \textrm{dim}(\Gamma _m)) (\textrm{dim}(\Gamma _{k-m})- \textrm{deg}(y))} \,\\{} & {} (p_{L,m})_! (x) \,\times \, (p_{L,k-m})_! (y) \end{aligned}$$

for homology classes \(x,y\in \textrm{H}_{\bullet }(SM)\). Noting that

$$\begin{aligned} \textrm{dim}(SM) + \textrm{dim}(\Gamma _m) = 2(N-1) + \lambda _m \end{aligned}$$

is odd for all \(m\in \mathbb {N}\) and that \(\textrm{dim}(\Gamma _{k-m})\) is even for all \(m,k\in \mathbb {N}\), \(m<k\) we can figure out the sign. The computation of \(\vee B_k^i\) is analogous. \(\square \)

Remark 7.6

Note at this point that it is sufficient to consider the critical manifolds with respect to the length function \({\mathcal {L}}_2\) of the form

$$\begin{aligned} \Sigma _1\times _M \Sigma _{k-1}, \ldots , \Sigma _{k-1}\times _M \Sigma _1 \subseteq \Lambda M\times _M \Lambda M. \end{aligned}$$

There are two other connected components at level kl, namely \(M\times _M \Sigma _k\) and \(\Sigma _k\times M\). However, as we saw now the corresponding homology classes in \(\textrm{H}_{\bullet }(\Lambda M\times _M\Lambda M)\) do not show up in the process of computing the coproduct, therefore we did not consider these components.

8 The cohomology product

In this section we briefly describe the Goresky-Hingston product on \(\mathbb {K}P^n\). We take cohomology with rational coefficients. Define classes

$$\begin{aligned} \sigma _k^i \in \textrm{H}^{\bullet }(\Lambda M,M) \quad \text {and} \quad \mu _k^i \in \textrm{H}^{\bullet }(\Lambda M,M) \end{aligned}$$

as follows. For \(k\in \mathbb {N}\) and \(i\in \{0,\ldots ,n-1\}\) the class \(\sigma _k^i\) is defined to be the dual of \(A_k^i\) and the class \(\mu _k^i\) is defined to be the dual of \(B_k^i\). Consequently,

$$\begin{aligned} \textrm{deg}(\sigma _k^i) = \lambda _k + i(\lambda +1 ) \quad \text {and} \quad \textrm{deg}(\mu _k^i) = \lambda _k + (i+1)(\lambda +1) + N-1. \end{aligned}$$

In particular, note that \(\textrm{deg}(\sigma _k^i)\) is odd and \(\textrm{deg}(\mu _k^i)\) is even for all \(k\in \mathbb {N}\), \(i\in \{0,\ldots ,n-1\}\). It is clear that the set

$$\begin{aligned} \{\sigma _k^i \,|\, k\in \mathbb {N}, i\in \{0,\ldots ,n-1\}\} \,\, \cup \,\, \{\mu _k^i \,|\, k\in \mathbb {N}, i\in \{0,\ldots ,n-1\}\} \end{aligned}$$

generates the cohomology \(\textrm{H}^{\bullet }(\Lambda M,M)\) additively. Using Theorem 7.5 we see that

and

for \(k,l\in \mathbb {N}\) and \(i,j\in \{0,\ldots ,n-1\}\). Moreover, we have

We want to mention at this point that the Goresky-Hingston product satisfies the following commutativity property. If \(x\in \textrm{H}^i(\Lambda M,M)\) and \(y\in \textrm{H}^j(\Lambda M,M)\), then

see [9, Theorem 2.14]. In particular, we get

which is consistent with the signs in Theorem 7.5.

Theorem 8.1

Let \(\mathbb {K}\) be \(\mathbb {C}\) or \(\mathbb {H}\) and consider the projective space \(M = \mathbb {K}P^n\). The Goresky-Hingston ring is multiplicatively generated by the classes

$$\begin{aligned} \sigma _1^0,\ldots ,\sigma _1^{n-1} \quad \text {and} \quad \mu _1^0,\ldots , \mu _1^{n-1} \end{aligned}$$

whose products are subject to the above relations. In particular, the ring is finitely generated and the element \(\sigma _1^0\) is non-nilpotent.

Remark 8.2

The existence of a non-nilpotent element is already shown in [7, 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 [7].