2. String topology product and coproduct

In this section we introduce the string topology operations, which we examine in this article. We follow [Reference Goresky and Hingston12] for the definition of the Chas-Sullivan product and [Reference Hingston and Wahl18] for the definition of the string topology coproduct.

Let $(M,g)$ be a closed Riemannian manifold of dimension $n$. We denote the unit interval by $I = [0,1]$. We define the path space of $M$ to be

\begin{equation*} PM = \big\{\gamma : I\to M\,|\, \gamma \,\,\text{absolutely continuous}, \,\, \int_0^1 g_{\gamma(t)}(\dot{\gamma}(t),\dot{\gamma}(t)) \,\mathrm{d}t \lt \infty \big\} \, .\end{equation*}

We refer to [Reference Klingenberg21, Definition 2.3.1] for the definition of absolute continuity of curves in a smooth manifold. It turns out that the path space $PM$ can be given the structure of a Hilbert manifold, see [Reference Klingenberg21, Section 2.3]. The free loop space of $M$ is defined to be the subspace

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

which is in fact a submanifold of $PM$. The underlying manifold $M$ can be seen as a submanifold of $\Lambda M$ via the identification with the trivial loops, see [Reference Klingenberg20, Proposition 1.4.6]. We consider the energy functional

\begin{equation*} E': PM \to [0,\infty),\qquad E'(\gamma) = \frac{1}{2}\int_0^1 g_{\gamma(t)}(\dot{\gamma}(t),\dot{\gamma}(t)) \mathrm{d}t \, . \end{equation*}

This is a differentiable function on the path space $PM$, see [Reference Klingenberg21, Theorem 2.3.20]. By restriction we obtain the energy functional on the free loop space, which we will denote by $E = E'|_{\Lambda M}$. The critical points of $E$ on $\Lambda M\smallsetminus M$ are precisely the closed geodesics in $M$. Moreover, for a point $p_0\in M$ we define

\begin{equation*} \Omega_{p_0}M = \{\gamma\in \Lambda M\,|\, \gamma(0) = p_0\} \end{equation*}

and call this space the based loop space of $M$ at the point $p_0$. The based loop space is a submanifold of the free loop space $\Lambda M$. Note that we will also frequently write $\Lambda$ for the free loop space $\Lambda M$ and $\Omega M$ or $\Omega$ for the based loop space if the manifold or the basepoint, respectively, are fixed. In the following let $R$ be a commutative ring with unit and consider homology and cohomology with coefficients in $R$. We assume that the manifold $M$ is oriented with respect to the coefficient ring $R$. If $\epsilon \gt 0$ is smaller than the injectivity radius of $M$, then the diagonal $\Delta M\subseteq M\times M$ has a tubular neighborhood

\begin{equation*} U_M := \{ (p,q)\in M\times M\,|\, \mathrm{d}(p,q) \lt \epsilon\} \, . \end{equation*}

Here, $\mathrm{d}$ is the distance function on $M$ induced by the Riemannian metric. Note that the normal bundle of $M\cong \Delta M$ in the product $M\times M$ is isomorphic to the tangent bundle $TM$. Consequently, since $U_M$ is a tubular neighborhood there is a homeomorphism of pairs

\begin{equation*} (U_M, U_M\smallsetminus M) \cong (TM,TM\smallsetminus M)\, . \end{equation*}

If we pull back the Thom class of $(TM,TM\smallsetminus M)$ via this homeomorphism we obtain a class

\begin{equation*} \tau_M\in \mathrm{H}^n(U_M,U_M\smallsetminus M) \, . \end{equation*}

On the free loop space $\Lambda M$ we consider the evaluation map $\mathrm{ev}_0\colon \Lambda\to M$ given by

\begin{equation*} \mathrm{ev}_0(\gamma) = \gamma(0) \, . \end{equation*}

We define the figure-eight space $\Lambda\times_M \Lambda$ as the pull-back

\begin{equation*} \Lambda\times_M\Lambda = (\mathrm{ev}_0\times \mathrm{ev}_0)^{-1}(\Delta M) = \{(\gamma,\sigma)\in \Lambda\times \Lambda\,|\,\gamma(0) = \sigma(0)\}\, . \end{equation*}

It turns out that the pull-back of the tubular neighborhood of the diagonal $U_M$ yields a tubular neighborhood of the figure-eight space

\begin{equation*} U_{CS} := (\mathrm{ev}_0\times\mathrm{ev}_0)^{-1}(U_M) = \{(\gamma,\sigma)\in\Lambda\times\Lambda\,|\,\mathrm{d}(\gamma(0),\sigma(0)) \lt \epsilon\} \, , \end{equation*}

see e.g. [Reference Hingston and Wahl18]. One can now pull back the Thom class $\tau_M$ via the map $\mathrm{ev}_0\times \mathrm{ev}_0$ to obtain a class

\begin{equation*} \tau_{CS} := (\mathrm{ev}_0\times\mathrm{ev}_0)^* \tau_M \in \mathrm{H}^n(U_{CS},U_{CS}\smallsetminus \Lambda\times_M \Lambda) \, . \end{equation*}

Let $R\colon U_{CS}\to \Lambda\times_M\Lambda$ be the retraction of the tubular neighborhood. Furthermore, for a time $s\in(0,1)$ there is a concatenation map

\begin{equation*} \mathrm{concat}_s\colon \Lambda\times_M\Lambda\to \Lambda \end{equation*}

given by

\begin{equation*} \mathrm{concat}_s(\gamma,\sigma)(t) := \begin{cases} \gamma(\tfrac{t}{s}), & 0\leq t\leq s \\ \sigma(\tfrac{t-s}{1-s}), & s \leq t \leq 1 \, . \end{cases} \end{equation*}

Note that for $s,t\in(0,1)$ the maps $\mathrm{concat}_{s}$ and $\mathrm{conact}_{t}$ are homotopic via re-parametrization. Thus, for the induced map in homology, we just write $\mathrm{concat}_*$. We can now define the Chas-Sullivan product.

Definition 2.1. Let $M$ be a closed $R$-oriented $n$-dimensional manifold. The Chas-Sullivan product is defined as the composition

\begin{eqnarray*} \wedge\colon \mathrm{H}_{i}(\Lambda)\otimes \mathrm{H}_j(\Lambda) &\xrightarrow{ (-1)^{n(n-i)}\times }& \mathrm{H}_{i+j}(\Lambda\times \Lambda) \\ &\xrightarrow{\qquad\qquad} & \mathrm{H}_{i+j}(\Lambda\times \Lambda, \Lambda\times \Lambda\smallsetminus \Lambda\times_M \Lambda) \\ &\xrightarrow{\ \ \text{excision}\ \ }& \mathrm{H}_{i+j}(U_{CS}, U_{CS}\smallsetminus \Lambda\times_M\Lambda) \\ &\xrightarrow{\quad\ \tau_{CS}\cap\ \ }& \mathrm{H}_{i+j-n}(U_{CS}) \\ &\xrightarrow[]{\qquad R_*\quad} & \mathrm{H}_{i+j-n}(\Lambda\times_M \Lambda) \\ & \xrightarrow[]{\ \ \mathrm{concat}_*\ \ } & \mathrm{H}_{i+j-n}(\Lambda) \, . \end{eqnarray*}

We now turn to the definition of the string topology coproduct. Recall that we had the explicit tubular neighborhood of the diagonal $U_M$. Let $\epsilon_0 \gt 0$ be a small positive number with $\epsilon_0 \lt \epsilon$ and define

\begin{equation*} U_{M,\geq \epsilon_0} = \{(p,q)\in U_M\,|\, \mathrm{d}(p,q)\geq \epsilon_0\}\, . \end{equation*}

Note that by [Reference Hingston and Wahl18] the pair $(U_M,U_{M,\geq\epsilon_0})$ is homeomorphic to $(TM^{ \lt \epsilon}, TM^{ \lt \epsilon}_{\geq\epsilon_0})$, where

\begin{equation*} TM^{ \lt \epsilon} := \{v\in TM\,|\, |v| \lt \epsilon\} \quad \text{and}\quad TM^{ \lt \epsilon}_{\geq\epsilon_0} := \{v\in TM^{ \lt \epsilon}\,|\, |v|\geq \epsilon_0\} \end{equation*}

and where $|\cdot|$ is the fiberwise norm induced by the Riemannian metric on $M$. Moreover, by [Reference Hingston and Wahl18] the Thom class in $(TM,TM\smallsetminus M)$ induces a Thom class in $(TM^{\epsilon},TM^{ \lt \epsilon}_{\geq\epsilon_0})$. Consequently, we obtain a class $\tau_M'\in \mathrm{H}^n(U_M,U_{M,\geq\epsilon_0})$. Fix a basepoint $p_0\in M$ and consider the sets

\begin{equation*} B_{p_0} := \{q\in M\,|\, \mathrm{d}(p_0,q) \lt \epsilon\}\quad \text{and}\quad B_{p_0,\geq\epsilon_0} := \{q\in B_{p_0}\,|\, \mathrm{d}(p_0,q)\geq \epsilon_0\} \, .\end{equation*}

Let $i$ be the inclusion

\begin{equation*} i\colon B_{p_0} \xrightarrow[]{\cong} \{p_0\}\times B_{p_0} \hookrightarrow U_M\, . \end{equation*}

This induces a map of pairs $j\colon (B_{p_0},B_{p_0,\geq\epsilon_0})\to (U_M,U_{M,\geq\epsilon_0})$. If we pull back the class $\tau_M'$ via this map we obtain a class

\begin{equation*} \tau_{p_0} := j^*\tau_M' \in \mathrm{H}^n(B_{p_0},B_{p_0,\geq\epsilon_0}) \cong \mathrm{H}(M,M\smallsetminus\{p_0\}) \end{equation*}

and by the properties of a Thom class this class coincides with the generator of $\mathrm{H}(M,M\smallsetminus\{p_0\})$ induced by the orientation of $M$.

On the product of the free loop space with the unit interval $\Lambda\times I$ there is an evaluation map

\begin{equation*} \mathrm{ev}_I\colon \Lambda\times I\to M\times M \end{equation*}

given by

\begin{equation*} \mathrm{ev}_I(\gamma,s) = (\gamma(0),\gamma(s)) \, . \end{equation*}

Define

\begin{equation*} U_{GH} = (\mathrm{ev}_I)^{-1}(U_M) = \{(\gamma,s) \in \Lambda M\times I\,|\, \mathrm{d}(\gamma(0),\gamma(s)) \lt \epsilon\} \end{equation*}

and

\begin{equation*} U_{GH,\geq\epsilon_0} = (\mathrm{ev}_I)^{-1}(U_{M,\geq\epsilon_0}) = \{(\gamma,s)\in U_{GH}\,|\, \mathrm{d}(\gamma(0),\gamma(s))\geq \epsilon_0\}\, . \end{equation*}

The set $U_{GH}$ is an open neighborhood of the subspace

\begin{equation*} F_{GH} := (\mathrm{ev}_I)^{-1}(\Delta M) = \{(\gamma,s)\in \Lambda\times I\,|\, \gamma(0)=\gamma(s)\}\, . \end{equation*}

The evaluation $\mathrm{ev}_I$ defines a map of pairs

\begin{equation*} \mathrm{ev}_I\colon (U_{GH} ,U_{GH,\geq\epsilon_0}) \to (U_M,U_{M,\geq\epsilon_0}) \end{equation*}

and we can pull back the class $\tau_M'$ to obtain a class

\begin{equation*} \tau_{GH} := (\mathrm{ev}_I)^* \tau_M'\in \mathrm{H}^n( U_{GH} ,U_{GH,\geq\epsilon_0})\, . \end{equation*}

The above constructions can be done similarly for the based loop space, i.e. we obtain an open neighborhood $ U^{\Omega}_{GH} = U_{GH}\cap (\Omega\times I) $ of the space

\begin{equation*} F^{\Omega}_{GH} := F_{GH}\cap (\Omega\times I) = \{(\gamma,s)\in \Omega\times I\,|\, \gamma(s) = p_0\} \end{equation*}

and we pull back the class $\tau_{p_0}$ to a class

\begin{equation*} \tau^{\Omega}_{GH} \in \mathrm{H}^n(U^{\Omega}_{GH},U^{\Omega}_{GH,\geq\epsilon_0}) \end{equation*}

where $U^{\Omega}_{GH,\geq\epsilon_0} := U_{GH,\geq\epsilon_0} \cap (\Omega\times I)$.

As Hingston and Wahl argue in [Reference Hingston and Wahl18], there is a retraction map $R_{GH}\colon U_{GH}\to F_{GH}$ and it is easy to see that this restricts to a retraction $U_{GH}^{\Omega}\to F^{\Omega}_{GH}$. Furthermore, we have a cutting map $\mathrm{cut}\colon F_{GH}\to \Lambda\times \Lambda$ given by

\begin{equation*} \mathrm{cut}(\gamma,s) = (\gamma|_{[0,s]},\gamma|_{[s,1]}) , \end{equation*}

where by $\gamma|_{[0,s]}$ we mean a reparametrized version of the restriction $\gamma|_{[0,s]}$. Again, this restricts appropriately to a based version.

Let $[I]$ be the positively oriented generator of $\mathrm{H}_1(I,\partial I)$ with respect to the standard orientation of the unit interval. We write $p_0$ for the basepoint of the based loop space which is just the trivial loop at the point $p_0$ and by abuse of notation we shall denote the set consisting of the single element $p_0$ just by $p_0$.

Remark 2.4. If we take $R=\mathbb{F}$ to be a field, then by the Künneth isomorphism the string topology coproduct induces a map

\begin{equation*} \vee' \colon \mathrm{H}_i(\Lambda,M) \to (\mathrm{H}_{*}(\Lambda,M)\otimes \mathrm{H}_{*}(\Lambda,M))_{i+1-n} \, . \end{equation*}

After a sign correction this map is indeed coassociative and graded cocommutative, see [Reference Hingston and Wahl18, Theorem 2.14] and therefore justifies the name “coproduct”.

Note that the based string topology coproduct and the string topology coproduct on the free loop space are compatible with respect to the inclusion $i\colon \Omega \hookrightarrow \Lambda$. More precisely, we have the commutativity of the following diagram, see [Reference Stegemeyer31, Proposition 2.4].

By dualizing, the coproduct with coefficients in a field induces a product in cohomology.

Definition 2.5. Let $\mathbb{F}$ be a field and take homology and cohomology with coefficients in $\mathbb{F}$. Let $M$ be a closed $\mathbb{F}$-oriented manifold. If $\alpha,\beta\in \mathrm{H}^{*}(\Lambda M,M;\mathbb{F})$ are relative cohomology classes, then the Goresky-Hingston product is defined by taking the dual to the string topology coproduct, i.e.

Here, we make the canonical identification $\mathrm{H}^{*}(\Lambda M,M;\mathbb{F}) \cong \mathrm{Hom}(\mathrm{H}_{*}(\Lambda M,M;\mathbb{F}),\mathbb{F})$.

Note that the Goresky-Hingston product can also be defined for more general coefficients, see [Reference Hingston and Wahl18, Section 1.6].

We conclude this section by defining the basepoint intersection multiplicity of a relative homology class for both the based loop space and the free loop space. Hingston and Wahl define the notion of basepoint intersection multiplicity in [Reference Hingston and Wahl18, Section 5] only in the case of the free loop space. The analogous concept for the based loop space is straight-forward.

Note that in the present paper loops are maps $\gamma\colon [0,1]\to M$ and therefore any loop has a basepoint intersection at $t=0$ and $t= 1$. We therefore subtract $1$ in the above definition in order not to count this trivial intersection twice. In [Reference Hingston and Wahl18] loops are considered as maps $\mathbb{S}^1\to M$ and therefore the additional term $-1$ does not show up in [Reference Hingston and Wahl18].

By considering finite-dimensional models of $\Lambda M$, respectively of $\Omega_{p_0}M$, one can see that the intersection multiplicity of a homology class is finite. Hingston and Wahl prove in the case of the free loop space that if a homology class $Y\in \mathrm{H}_{*}(\Lambda M,M)$ has intersection multiplicity $1$ then its coproduct vanishes, see [Reference Hingston and Wahl18, Theorem 3.10]. Since the proof of this theorem which includes Lemma 3.5 and Proposition 3.6 of [Reference Hingston and Wahl18] only uses maps that keep the basepoint of the loops fixed, the proof can be transferred directly to the case of the based loop space. We obtain the following result.

3. Completing manifolds

In this section we introduce the concept of a completing manifold. This will be the key technique in our study of the string topology operations on symmetric spaces in Sections 6 and 5. We closely follow [Reference Hingston and Oancea16] and [Reference Oancea28].

Assume that $X$ is a Hilbert manifold and that $F \colon X\to \mathbb{R}$ is a Morse-Bott function. Let $a$ be a critical value of $F$ and assume that the set of critical points at level $a$ is a non-degenerate submanifold $\Sigma$ with index $\lambda$ and with orientable negative bundle. As usual in Morse theory we denote by $X^{\leq a}$ the set

\begin{equation*} X^{\leq a} := \{x\in X\,|\, F(x)\leq a\}\, . \end{equation*}

Similarly, one defines the strict sublevel set $X^{ \lt a}$. It is well-known that the level homology $\mathrm{H}_{*}(X^{\leq a},X^{ \lt a})$ is completely determined by the homology of the critical manifold $\Sigma$. More precisely, we have

\begin{equation*} \mathrm{H}_{*}(X^{\leq a},X^{ \lt a}) \cong \mathrm{H}_{* - \lambda}(\Sigma) \, . \end{equation*}

In general, there is no direct link between the homology of $\Sigma$ and the homology of the sublevel set $X^{\leq a}$. As we shall see in the following the existence of a completing manifold however is sufficient for the property that the homology of $X^{\leq a}$ is a direct sum of the homology of the strict sublevel set $X^{ \lt a}$ and the homology of the critical manifold $\Sigma$.

Definition 3.1. Let $X$ be a Hilbert manifold and let $F\colon X\to \mathbb{R}$ be a Morse-Bott function. Suppose that $a$ is a critical value of $F$ and assume that $\Sigma$ is the orientable non-degenerate critical submanifold at level $a$ of index $\lambda$ and of dimension $m = \mathrm{dim}(\Sigma)$. A completing manifold for $\Sigma$ is a closed, orientable manifold $\Gamma$ of dimension $\lambda + m$ with a closed submanifold $s\colon S\hookrightarrow \Gamma$ of codimension $\lambda$ and a map $f: \Gamma \to X^{\leq a}$ such that the following holds. The map $f$ is an embedding near $S$ with the restriction $f|_S$ mapping $S$ homeomorphically onto $\Sigma$ and such that $f^{-1}(\Sigma) = S$. Furthermore, there is a retraction map $p:\Gamma\to S$, i.e. $p \circ s = \mathrm{id}_S$, and the map $f$ induces a map of pairs

\begin{equation*} f \colon (\Gamma,\Gamma\smallsetminus S) \to (X^{\leq a}, X^{ \lt a}) \, . \end{equation*}

In case that we consider homology with $\mathbb{Z}_2$-coefficients we drop the orientability conditions on $\Sigma$ and $\Gamma$.

Let $g \colon M\to N$ be a map between closed oriented manifolds. We recall that the Gysin map

\begin{equation*} g_!\colon \mathrm{H}_i(N) \to \mathrm{H}_{i+\mathrm{dim}(M)-\mathrm{dim}(N)}(M)\end{equation*}

is defined by

\begin{align*}g_! \colon \mathrm{H }_i(N) & \xrightarrow[]{(PD_{N})^{-1}} \mathrm{H}^{\mathrm{dim}(N) - i}(N) \xrightarrow[]{g^*} \mathrm{H}^{\mathrm{dim}(N)-i}(M) \\ &\xrightarrow[]{PD_{M}} \mathrm{H}_{\mathrm{dim}(M) - (\mathrm{dim}(N)-i)}(M) , \end{align*}

where

\begin{equation*}\mathrm{PD}_L\colon \mathrm{H}^j(L) \xrightarrow[]{\cong} \mathrm{H}_{\mathrm{dim}(L)-j}(L) \end{equation*}

stands for Poincaré duality on the closed oriented manifold $L$. In particular, we note that Gysin maps behave contravariantly, i.e. if $h\colon N\to L$ is a second map between compact oriented manifolds, then

(3.1)\begin{equation} (h\circ g)_! = g_!\circ h_! \, . \end{equation}

If we use $\mathbb{Z}_2$-coefficients then all orientability conditions in the above can be dropped.

Let $X$ be a Hilbert manifold with a Morse-Bott function $F\colon X\to \mathbb{R}$. Assume that $\Sigma$ is an oriented critical submanifold at level $a$ and that $\Gamma$ is an oriented completing manifold for $\Sigma$. Then we have the embedded submanifold $s\colon S\hookrightarrow \Gamma$ and by definition there is retraction map $p\colon \Gamma\to S$, i.e. we have

(3.2)\begin{equation} p\circ s = \operatorname{id}_S\, . \end{equation}

Moreover, note that we get a map

\begin{equation*} \mathrm{H}_i(\Gamma)\to \mathrm{H}_i(\Gamma,\Gamma\smallsetminus S)\xrightarrow[]{\cong} \mathrm{H}_{i-\lambda}(S) \end{equation*}

where the first map is induced by the inclusion of pairs and the second one is the composition of excision and the Thom isomorphism. It turns out, see [Reference Bredon3, Theorem 11.3], that this map coincides up to sign with the Gysin map $s_!$. Consequently, by equations (3.1) and (3.2) we see that $p_!$ is a right inverse for this map up to sign. In particular, the long exact sequence of the pair $(\Gamma,\Gamma\smallsetminus S)$ yields a short exact sequence in each degree, which splits via the map $p_!$. Thus, for each $i\in \mathbb{N}$ we have a short exact sequence

\begin{equation*} 0 \to \mathrm{H}_{i}(\Gamma\smallsetminus S) \to \mathrm{H}_i(\Gamma) \xrightarrow[]{s_!} \mathrm{H}_{i-\lambda}(S) \to 0 \end{equation*}

and due to the splitting $p_!$ we see that

\begin{equation*} \mathrm{H}_i(\Gamma) \cong \mathrm{H}_{i}(\Gamma\smallsetminus S)\oplus \mathrm{H}_{i-\lambda}(S) \cong \mathrm{ker}(s_!)\oplus \mathrm{H}_{i-\lambda}(S)\cong\mathrm{ker}(s_!)\oplus \mathrm{im}(p_!) \, . \end{equation*}

Combining these ideas with the existence of the map $f\colon \Gamma\to X$ one can show the following.

Proposition 3.3. ([Reference Oancea28], Lemma 6.2)

Let $X$ be a Hilbert manifold, $F\colon X\to \mathbb{R}$ a Morse-Bott function and let $a$ be a critical value of $F$. Assume that the set of critical points at level $a$ is an oriented non-degenerate critical submanifold $\Sigma$ of index $\lambda$. If there is a completing manifold for $\Sigma$ then

(3.3)\begin{equation} \mathrm{H}_{*}(X^{\leq a}) \cong \mathrm{H}_{*}(X^{ \lt a}) \oplus \mathrm{H}_{* }(X^{\leq a},X^{ \lt a}) \cong \mathrm{H}_{*}(X^{ \lt a}) \oplus \mathrm{H}_{* -\lambda}(\Sigma) \, . \end{equation}

Remark 3.4. Note that the construction of a completing manifold does not only give us the isomorphism of Proposition 3.3, but it also yields an explicit way of describing the homology classes which come from the critical manifold $\Sigma$. More precisely, by the above discussion it is clear that the composition

\begin{equation*} f_*\circ p_! \colon \mathrm{H}_i(\Sigma)\to \mathrm{H}_{i+\lambda}(\Gamma)\to \mathrm{H}_{i+\lambda}(X^{\leq a}) \end{equation*}

is injective. Here, the map $f$ is the map of the completing manifold $\Gamma$ into $X$ and $p$ is the retraction $\Gamma\to S\cong \Sigma$.

Recall that the homology of the completing manifold $\Gamma$ splits as

\begin{equation*} \mathrm{H}_i(\Gamma)\cong \mathrm{H}_i(\Gamma\smallsetminus S)\oplus \mathrm{H}_{i}(\Gamma,\Gamma\smallsetminus S) \cong \mathrm{H}_i(\Gamma\smallsetminus S)\oplus \mathrm{H}_{i-\lambda}(\Sigma)\, . \end{equation*}

We note that, while we usually only care about the classes in $\mathrm{H}_i(\Gamma)$ that are in the image of $p_!\colon \mathrm{H}_i(\Sigma)\to \mathrm{H}_{i+\lambda}(\Gamma)$, we cannot say much about the image of $\mathrm{H}_i(\Gamma\smallsetminus S)$ under the map $f_*$. For example, it is not clear in general when a class in $\mathrm{H}_i(\Gamma\smallsetminus S)$ lies in the kernel of $f_*$. However, by the commutativity of the diagram

we do see that $\mathrm{H}_i(\Gamma\smallsetminus S)$ is mapped to the homology of the strict sublevel set $\mathrm{H}_i(X^{ \lt a})$ via $f_*$.

If $X$ is a Hilbert manifold with a Morse-Bott function $F\colon X\to \mathbb{R}$ let us assume that for every critical value $a$ we have

(3.4)\begin{equation} \mathrm{H}_{*}(X^{\leq a}) \cong \mathrm{H}_{*}(X^{ \lt a}) \oplus \mathrm{H}_{*}(X^{\leq a},X^{ \lt a})\, . \end{equation}

In this case we say that the function $F$ is a perfect Morse-Bott function. This property clearly holds if every critical submanifold is oriented and has a completing manifold. Ziller shows in [Reference Ziller35] that the energy functional on the free loop space of a compact symmetric space is perfect if one takes $\mathbb{Z}_2$-coefficients. He uses explicit completing manifolds, which we shall study in detail in Section 5 in order to compute the Chas-Sullivan product on symmetric spaces partially.

4. Geodesics in symmetric spaces

In this section we review some basic constructions in the theory of globally symmetric spaces. In Sections 5 and 6 we shall use the concepts from this section and therefore want to introduce the necessary tools in a concise manner. A general reference for the geometry of symmetric spaces is [Reference Helgason14].

A connected Riemannian manifold $(M,g)$ is a symmetric space if for each point $p\in M$ there is an isometry $s_p\colon M\to M$ that fixes $p$ and such that $(\mathrm{D}s_p)_p = -\mathrm{id}_{T_p M}$. The isometry $s_p$ is also called geodesic involution at $p$. Note that the isometry group of any closed Riemannian manifold is a compact Lie group. Let $G$ be the connected component of the identity of the group of isometries of the symmetric space $(M,g)$ and fix a basepoint $o\in M$. There is an involutive Lie group automorphism $S\colon G\to G$ which is given by $S(\varphi) = s_o\circ \varphi\circ s_o$. Since we assume that $M$ is connected, the group $G$ acts transitively on $M$, see [Reference Helgason14, Theorem IV.3.3]. Hence, $M$ is a homogeneous space $M = G/K$ with $K$ being the stabilizer of $o$ under the action of $G$. It further holds that

\begin{equation*} \mathrm{Fix}(S)_0 \subseteq K\subseteq \mathrm{Fix}(S) \, . \end{equation*}

Here, $\mathrm{Fix}(S)$ denotes the subgroup of $G$ consisting of elements which are fixed by the involutive automorphism $S$, i.e.

\begin{equation*} \mathrm{Fix}(S) = \{k\in G\,|\,S(k) = k\} \, . \end{equation*}

Moreover, by the subscript $0$ we denote the connected component of the unit element $e\in G$.

Let $\mathfrak{g} \cong T_e G$ be the Lie algebra of $G$. The differential $\mathrm{D}S_e\colon \mathfrak{g}\to \mathfrak{g}$ is an involutive Lie algebra automorphism of $\mathfrak{g}$. Therefore, $\mathfrak{g}$ has a decomposition of the form

\begin{equation*} \mathfrak{g} = \mathfrak{k}\oplus \mathfrak{m}\, , \end{equation*}

with

\begin{equation*} \mathfrak{k} = \mathrm{Eig}(\mathrm{D}S_e\,,+1) \quad \text{and}\quad \mathfrak{m} = \mathrm{Eig}(\mathrm{D}S_e\,,-1)\, \end{equation*}

where $\mathrm{Eig}(DS_e,\pm 1)$ is the $\pm 1$-eigenspace in $\mathfrak{g}$ of the linear map $\mathrm{D}S_e$. The subspace $\mathfrak{k}$ is precisely the Lie algebra of the subgroup $K$ and sits inside $\mathfrak{g}$ as a Lie subalgebra, see [Reference Helgason14, Theorem IV.3.3]. Moreover, the Lie bracket of $\mathfrak{g}$ restricted to $\mathfrak{m}$ satisfies $[\mathfrak{m},\mathfrak{m}]\subseteq \mathfrak{k}$. The adjoint action $\mathrm{Ad}\colon G\times \mathfrak{g}\to \mathfrak{g}$ of $G$ restricts to an action of $K$ on $\mathfrak{m}$. To see this, let $k\in K$ and $g\in G$. Since $K\subseteq \mathrm{Fix}(S)$ we have that

\begin{equation*} S (kgk^{-1}) = s_o kgk^{-1} s_o = s_o k s_o s_o g s_o s_o k^{-1} s_o = S(k) s_o g s_o S(k^{-1}) = k S(g) k^{-1} . \end{equation*}

Hence, it holds that $S\circ \mathrm{Conj}_k = \mathrm{Conj}_k \circ S$ and by differentiating this equation at $e\in G$ one checks that the eigenspaces of $\mathrm{D}S_e$ are preserved by $(\mathrm{D}\mathrm{Conj}_k )_e = \mathrm{Ad}_k\colon \mathfrak{g}\to \mathfrak{g}$. Note that if $\pi\colon G\to M = G/K$ is the canonical projection then

\begin{equation*} \mathrm{D}\pi_e |_{\mathfrak{m}} \colon \mathfrak{m} \to T_o M \end{equation*}

is an isomorphism, see [Reference Helgason14, Theorem IV.3.3]. Moreover, the action of $K$ on $T_o M$ which is induced by differentiating the action of $K$ on $M$ is equivalent to the adjoint action of $K$ on $\mathfrak{m}$ via $\mathrm{D}\pi_e|_{\mathfrak{m}}$.

Next, we discuss the roots of a compact symmetric space. Since the structure theory of Lie algebras behaves particularly well for complex Lie algebras we consider the complexification $\mathfrak{g}_{\mathbb{C}}$ of $\mathfrak{g}$ for a moment. One introduces the notion of a Cartan subalgebra $\mathfrak{h}\subseteq \mathfrak{g}_{\mathbb{C}}$ of $\mathfrak{g}_{\mathbb{C}}$, see [Reference Helgason14, p. 162] for a definition. This is in particular a maximal abelian subalgebra of $\mathfrak{g}_{\mathbb{C}}$. Note that the adjoint maps $\mathrm{ad}_{H_1},\mathrm{ad}_{H_2}\colon \mathfrak{g}\to \mathfrak{g}$ commute for $H_1,H_2\in\mathfrak{h}$ since $\mathfrak{h}$ is abelian and thus

\begin{equation*}\mathrm{ad}_{H_1}\circ \mathrm{ad}_{H_2} - \mathrm{ad}_{H_2}\circ \mathrm{ad}_{H_1} = \mathrm{ad}_{[H_1,H_2]} = 0 .\end{equation*}

Consequently, the maps $\mathrm{ad}_H\colon \mathfrak{g}\to \mathfrak{g}$, $H\in\mathfrak{h}$ can be diagonalized simultaneously. We want to control the eigenvalues of $\mathrm{ad}_H$ as $H$ varies in $\mathfrak{h}$ which leads to the notion of a root. A root of $\mathfrak{g}_{\mathbb{C}}$ is a non-zero element $\alpha$ of the dual space $\mathfrak{h}^*$ such that there is a non-trivial element $X\in\mathfrak{g}_{\mathbb{C}}$ with

\begin{equation*} \mathrm{ad}_H(X) = [H,X] =\pi \, i \, \alpha(H) X \quad \text{for all} \,\,\, H\in \mathfrak{h}\, . \end{equation*}

We denote the set of roots of $\mathfrak{g}_{\mathbb{C}}$ by $\Delta'$. By definition the root subspace

\begin{equation*} \mathfrak{g}^{\alpha} = \{X\in \mathfrak{g}_{\mathbb{C}} \,|\, [H,X] = \pi \,i\, \alpha(H) X \} \end{equation*}

is a non-trivial subspace of $\mathfrak{g}_{\mathbb{C}}$. It turns out that if $\mathfrak{g}_{\mathbb{C}}$ semisimple, then $\mathfrak{g}_{\mathbb{C}}$ decomposes as the direct sum of the Cartan subalgebra $\mathfrak{h}$ and the direct sum over all root subspaces $\mathfrak{g}^{\alpha}$, $\alpha\in \Delta'$, i.e.

\begin{equation*} \mathfrak{g}_{\mathbb{C}} = \mathfrak{h} \oplus \bigoplus_{\alpha\in\Delta'} \mathfrak{g}^{\alpha} , \end{equation*}

see [Reference Helgason14, Theorem III.4.2]. The set of roots therefore has to be finite.

Coming back to the geometry of the symmetric space $(M,g)$ note that the subspace $\mathfrak{m}\subseteq \mathfrak{g}$ which we had identified with the tangent space $T_o M$ via $\mathrm{D}\pi_e$ can also be seen as a subspace of the complexified Lie algebra $\mathfrak{m}\subseteq \mathfrak{g}\subseteq \mathfrak{g}_{\mathbb{C}}$. It turns out that there is a maximal abelian subspace $\mathfrak{a} \subseteq \mathfrak{m}$ of $\mathfrak{m}$ such that $\mathfrak{a}$ is contained in a Cartan subalgebra $\mathfrak{h}$ of $\mathfrak{g}_{\mathbb{C}}$, see [Reference Helgason14, p. 284]. Recall that $\mathfrak{m}$ is not a Lie algebra, but rather $[\mathfrak{m},\mathfrak{m}]\subseteq \mathfrak{k}$. By an abelian subspace of $\mathfrak{m}$ we mean in this case that $\mathfrak{a}$ satisfies $[\mathfrak{a},\mathfrak{a}]= \{0\}$. We call the subspace $\mathfrak{a}$ a Cartan subalgebra of $(G,K)$. Let $\Delta$ be the set of roots $\alpha\in \Delta'$ such that $\alpha|_{\mathfrak{a}} \neq 0$ where we identify two roots $\alpha_1,\alpha_2\in \Delta'$ if $\alpha_1|_{\mathfrak{a}} = \alpha_2|_{\mathfrak{a}}$. We say that $\Delta$ is the set of roots of $(M,g)$. We remark that the roots of the symmetric space $(M,g)$ are real-valued and they govern a lot of the geometric features of $M$. In particular the conjugate points along geodesics can be read off from the root system as we shall see later.

Further, we define the map $\mathrm{Exp}\colon \mathfrak{m}\to M$ by

\begin{equation*} \mathrm{Exp} = \pi\circ \mathrm{exp}|_{\mathfrak{m}} \end{equation*}

with $\mathrm{exp}$ being the Lie group exponential of the Lie group $G$. It turns out that this map is in fact the Riemannian exponential map at the point $p$ under the canonical identification $T_o M\cong \mathfrak{m}$, see [Reference Helgason14, Theorem IV.3.3]. Set $T = \mathrm{Exp}(\mathfrak{a})\subseteq M$. This is a flat totally geodesic and homogeneous submanifold and therefore isometric to an embedded flat torus. We call $T$ the maximal torus. In particular, the geodesics in $T$ are just the images of the straight lines in $\mathfrak{a}$ under the exponential map $\mathrm{Exp}$.

Before we move on to the behavior of geodesics in symmetric spaces, we want to introduce the notion of the rank of a symmetric space.

Note that the rank of $M$ does not depend on the choice of $\mathfrak{a}$, see [Reference Helgason14, Lemma V.6.3]. The compact symmetric spaces of rank $1$ are precisely the spheres and the real, complex and quaternionic projective spaces as well as the Cayley plane. Further examples of compact symmetric spaces are compact Lie groups; real, complex and quaternionic Grassmannians as well as Lagrange Grassmannians and the space of orthogonal complex structures in $\mathbb{R}^{2n}$. In fact symmetric spaces can be classified by extending the classification results for simple Lie algebras to the setting of symmetric spaces. See [Reference Helgason14, Chapter X] for details.

We now want to see what the roots of the compact symmetric space $(M,g)$ tell us about the geometry of $M$. Consider a pair $(\alpha,n)$ where $\alpha\in \Delta$ and $n\in \mathbb{Z}$. This defines an affine hyperplane

\begin{equation*} \{H\in \mathfrak{a}\,|\, \alpha(H) = n \}\subseteq \mathfrak{a} \end{equation*}

which we call a singular plane and which we denote by $(\alpha,n)$. As we have seen above, the geodesics in $T$ starting at $o$ are images of lines through the origin in $\mathfrak{a}$ under $\mathrm{Exp}$. Let

\begin{equation*}\sigma_H\colon [0,\infty)\to \mathfrak{a} ,\quad \sigma_H(t) = tH\end{equation*}

for some $H\in\mathfrak{a}$ be such a ray and let

\begin{equation*} \gamma_H\colon [0,\infty)\to M,\quad \gamma_H = \operatorname{Exp}\circ \, \sigma_H \end{equation*}

be the corresponding geodesic in $M$. Define times $0 = t_0 \lt t_1 \lt t_2 \lt \ldots $ inductively by setting $t_0 = 0$ and

\begin{equation*} t_i := \inf\{t\in (t_{i-1},\infty)\,|\, \sigma(t)\text{lies in a hyperplane }(\alpha,n),\,\,n\neq 0 \} \end{equation*}

for $i\geq 1$. Of course, it is also possible that $\sigma$ meets an intersection of several hyperplanes at a time $t_i$, $i\in \mathbb{N}$. The following proposition states that the singular planes encode the conjugate points in the symmetric space $M$.

We refer to Figure 1 for a sketch of the singular planes of a symmetric space of rank $2$. A priori Proposition 4.2 is only a statement about the geodesics in $M$ that start at $p$ and that lie in the maximal torus $T$. However, the following lemma shows that every geodesic $\gamma\colon[0,\infty)\to M$ is of the form $\gamma_H$ as above up to a global isometry. Note that the action of $G$ on the symmetric space $M$ induces an action $\Phi\colon G\times \Lambda M\to \Lambda M$ on the free loop space given by

\begin{equation*} \Phi(g,\gamma)(t) := g\cdot \gamma(t)\quad \text{for}\,\,g\in G,\gamma\in\Lambda M\, . \end{equation*}

Proof. Since the isometry group acts transitively on $M$, we find an element $g'\in G$ such that $\gamma_1 = g'\cdot \gamma\colon [0,\infty)\to M$ starts at the base point $o = \pi(e)\in M$. Let $H_1 = \dot{\gamma}_1(0)\in \mathfrak{m}$ be the tangent vector at $t= 0$. We have

\begin{equation*}\gamma_1(t) = \operatorname{Exp}(t\cdot H_1) = \pi\circ \exp(t \cdot H_1) . \end{equation*}

However, $H_1$ might not lie in $\mathfrak{a}$. By [Reference Helgason14, Theorem VII.8.6], we can find an element $k\in K$ such that $H := \mathrm{Ad}_k(H_1) \in \mathfrak{a}$. Consider the geodesic $\gamma_2\colon [0,\infty)\to M, t\mapsto k\cdot \gamma_1(t)$. We have

\begin{equation*} \gamma_2(t) = k\cdot \gamma_1(t) = \pi \big( k \exp(t\cdot H_1) \big) = \pi\big( k \exp(t\cdot H_1 ) k^{-1} \big) = \operatorname{Exp}(t\cdot \mathrm{Ad}_k(H_1)) = \gamma_H(t) \end{equation*}

for all $i\in [0,\infty)$. Hence, $\gamma_2 = \gamma_H$ and by construction we have $\gamma_2 = (kg')\cdot \gamma$. Setting $g = kg'$ yields the claim.

Figure 1. Example of the maximal abelian subspace of a symmetric space of rank $2$. The above figure shows the maximal abelian subspace for the complex Grassmannian $\mathrm{Gr}_2(\mathbb{C}^4)$, see [Reference Sakai29, Section 4.2] and [Reference Sakai30, Example 5.5]. In this example there are four positive roots $\{\alpha,\beta,\delta,\epsilon\}$ and the corresponding singular planes are drawn as dashed or dotted lines. The lattice points are pictured as dots. The ray $\sigma_H$ is mapped to a closed geodesic by $\operatorname{Exp}$ since it intersects the lattice $\mathcal{F}$ at the point $H$. Observe that there are five conjugate points in the interior of the corresponding closed geodesic $\gamma_H$. They can be read off by considering the intersections of $\sigma_H$ with the singular planes.

Since $\operatorname{Exp}\colon \mathfrak{a}\to \operatorname{Exp}(\mathfrak{a})$ is the universal covering of the torus $T= \operatorname{Exp}(\mathfrak{a})$, we obtain a lattice $\mathcal{F}\subseteq \mathfrak{a}$ by setting

\begin{equation*} \mathcal{F} = \{H\in\mathfrak{a}\,|\, \operatorname{Exp}(H) = o\} \, . \end{equation*}

Since the maximal torus $T$ is isometric to a flat torus, the closed geodesics in $T$ are in one-to-one correspondence to points in the lattice $\mathcal{F}$. See Figure 1 for a sketch of the lattice of a symmetric space of rank $2$.

By Lemma 4.3 we see that the critical submanifolds of the energy functional in $\Lambda M$ are the orbits of the closed geodesics in $T$ under the action of $G$. However, note that several distinct points in the lattice $\mathcal{F}$ can correspond to closed geodesics which lie in the same orbit under the action of $G$. This indeterminacy can be removed as follows.

Recall from above that we have introduced the set $\Delta$ of roots of $(M,g)$. One can define certain orderings on $\Delta$, and define the set of positive roots $\Delta_+\subseteq \Delta$ with respect to an ordering, see [Reference Helgason14, Section VII.2] for details. We fix such an ordering on $\Delta$. The positive Weyl chamber with respect to this ordering is defined as

\begin{equation*}W:= \{X \in \mathfrak{a} \, | \, \alpha(X) \gt 0 \text{ for all } \alpha \in \Delta_+ \}. \end{equation*}

The positive Weyl chamber is a convex subset of $\mathfrak{a}$. For other choices of orderings of $\Delta$ we obtain other convex subsets of $\mathfrak{a}$, which are also called Weyl chambers, but they are disjoint from the positive Weyl chamber. The Weyl group $W(G,K)\subseteq \mathrm{Gl}(\mathfrak{a})$ of $M = G/K$ is the group generated by the reflections $s_{\alpha}\colon \mathfrak{a}\to \mathfrak{a}$ about the hyperplanes

\begin{equation*} (\alpha,0) = \{H\in \mathfrak{a} \,|\, \alpha(H) = 0\} \, \end{equation*}

for all $\alpha\in \Delta$. The Weyl group is finite and it acts simply transitively on the set of Weyl chambers, see [Reference Helgason14, Proposition VII.2.12]. In order to get a one-to-one correspondence between points in the lattice $\mathcal{F}$ and the critical submanifolds we need to consider the intersection of the lattice $\mathcal{F}$ with the closure of the Weyl chamber $\overline{W}$.

For a sketch of the positive Weyl chamber and the lattice, see Figure 2. Recall that we call a closed geodesic $\gamma\in \Lambda M$ prime if it is not the iterate of another closed geodesic, i.e. if there is no $l\geq 2$ and a closed geodesic $\sigma\in \Lambda M$ with $\gamma = \sigma^l$. We say that a critical submanifold is prime if it consists only of prime closed geodesics. The next proposition says that there are many prime critical submanifolds for higher rank symmetric spaces.

Proof. The lattice $\mathcal{F}\subseteq \mathfrak{a}$ is generated by elements $X_1,\ldots, X_r$ where $r$ is the rank of $M$. Every element of the form

(4.1)\begin{equation} \lambda_1 X_1 + \ldots + \lambda_r X_r \quad \text{with at least one } \lambda_i = \pm 1 , \,\, i\in\{1,\ldots,r\} \end{equation}

corresponds to a prime closed geodesic. If $r\geq 2$ there are countably infinite such elements. Recall that all Weyl chambers are homeomorphic to one another and that the union of the closures of the finitely many Weyl chambers is all of $\mathfrak{a}$. Since the action of the Weyl group preserves the lattice $\mathcal{F}$, see [Reference Helgason14, Theorem VII.8.5], there must be countably infinite elements of the form (4.1) in each Weyl chamber.

Figure 2. Example of the positive Weyl chamber for the case of the maximal abelian subspace shown in Figure 1. The Weyl chamber is depicted as the shaded area. The lattice points are denoted as bullet points. Note that the closed geodesic $\gamma_H = \operatorname{Exp}\circ \sigma_H$ is prime, since it intersects no lattice points before reaching $H$. Further note that the closed geodesics $\gamma_{X_1}$ and $\gamma_{X_2}$ lie in the same critical manifold in $\Lambda M$ since $X_2$ is mapped to $X_1$ by the reflection about the hyperplane $(\epsilon,0)$.

In contrast, for a rank one symmetric space it is well-known that there is exactly one prime critical submanifold of $\Lambda M$.

We conclude this section by computing the index growth of the closed geodesics in the compact symmetric space $M$. We have seen above that conjugate points appear along a closed geodesic of the form $\gamma_H = \operatorname{Exp}\circ \,\sigma_H$ for $H\in\mathfrak{a}$ whenever the ray $\sigma_H$ intersects a singular plane. We shall now study an alternative characterization of the conjugate points along closed geodesics which will be used in the next section. Let $c$ be a closed geodesic in $M$ starting at the basepoint $o\in M$. Let $K_c$ be the closed subgroup of $K$ that keeps $c$ pointwise fixed, i.e.

\begin{equation*} K_c = \{k\in K\,|\, k . c(t) = c(t) \,\,\text{for all}\,\, t\in I\}\, . \end{equation*}

The conjugate points along $c$ can be characterized as follows. A point $q = c(t) \in M$ with $t\in I$ is a conjugate point along $c$ if and only if the stabilizer

\begin{equation*} K_q := \{k\in K\, |\, k . q = q\} \end{equation*}

of $q$ satisfies

\begin{equation*} \mathrm{dim}(K_q) \gt \mathrm{dim}(K_c)\, . \end{equation*}

This can be seen as follows. Let $\mathfrak{k}_q$ be the Lie algebra of $K_q$ and $\mathfrak{k}_c$ be the Lie algebra of $K_c$. The Lie algebra $\mathfrak{k}_c$ is a Lie subalgebra of $\mathfrak{k}_q$. Let $\xi\in\mathfrak{k}_q$ be an element such that $\xi\not\in \mathfrak{k}_c$ and consider the one-parameter group $\Xi\colon \mathbb{R}\to K_q, \Xi(s) = \exp(s\xi)$. The variation of the geodesic $c$ defined by

\begin{equation*}[0,t]\times (-\epsilon,\epsilon)\to M , \quad (\tau,s)\mapsto \exp(s\xi) \cdot c(\tau) \end{equation*}

is a non-trivial variation of $c$ through geodesics with fixed endpoints. It thus determines a non-trivial Jacobi field and hence $q = c(t)$ is conjugate to $o$ along $c$. Elaborating on this idea one finds that the difference $\mathrm{dim}(K_q) - \mathrm{dim}(K_c)$ is precisely the multiplicity of the conjugate point $q$, i.e. the dimension of the space of Jacobi fields along $c|_{[0,t]}$ that vanish at the endpoints, see [Reference Ziller35, p. 13-14].

Recall that the closed geodesics in $M$ are the critical points of the energy functional on $\Lambda M$. We want to express the Morse index of a closed geodesic $c\in \Lambda M$ through the geometric data at hand. One shows that in the case of $(M,g)$ being a symmetric space, the index of the closed geodesic $c$ is equal to the sum of the multiplicities of the conjugate points in the interior, see [Reference Ziller35, Theorem 4]. This means that we have

\begin{equation*} \mathrm{ind}(c) = \sum_{0 \lt t_i \lt 1} (\mathrm{dim}(K_{c(t_i)}) - \mathrm{dim}(K_c))\, ,\end{equation*}

where the times $t_i\in(0,1)$ are such that $c(t_i)$ is a conjugate point as above. Define the integer

\begin{equation*}\mu =\mathrm{dim}(K) -\mathrm{dim}(K_c)\,. \end{equation*}

If $c$ is the $k$-th iterate of a prime closed geodesic $\sigma$, then by a simple counting argument we obtain

(4.2)\begin{equation} \mathrm{ind}(c) = k \, \mathrm{ind}(\sigma) + (k-1) \mu \end{equation}

since the origin appears $(k-1)$ times in the interior of $c$ and clearly has stabilizer $K$.

We now consider the critical manifolds of the energy functional in $\Lambda M$. By Proposition 4.4 the critical manifolds are in one-to-one correspondence with the lattice points that intersect the closure of the positive Weyl chamber, i.e. with the set $\mathcal{F}\cap \overline{W}$. Note that for the connected component $\Sigma_c$ of the critical set of $E$ which contains the closed geodesic $c= \gamma_H$ for $H\in\mathcal{F}\cap \overline{W}$, the $G$-action on $\Lambda M$ induces a transitive $G$-action on $\Sigma_c$. This is again due to Proposition 4.4. The stabilizer group of $c$ under this action is the group $K_c$ that we introduced above. Consequently, the critical manifold $\Sigma_c$ is diffeomorphic to the homogeneous space $G/K_c$. Ziller has shown that $G/K_c$ is a non-degenerate critical submanifold of $E$, see [Reference Ziller35, Theorem 2]. Since this holds for each critical submanifold, $E$ is a Morse-Bott function. The nullity of the closed geodesic $c$, i.e. the dimension of the kernel of the Hessian of $E$ at $c$, is thus equal to the dimension of the homogeneous space $G/K_c$. Since the canonical map

\begin{equation*}G/K_c \to G/K,\quad g K_c \mapsto gK \end{equation*}

is a fiber bundle with typical fiber $K/K_c$, we see that

\begin{equation*} \mathrm{null}(c) = \mathrm{dim}(G/K) + \mathrm{dim}(K/K_c) = n + \mu \, . \end{equation*}

Consequently, the sum of index and nullity of the $k$-th iterate of a prime closed geodesic $\sigma$ is

(4.3)\begin{equation} (\mathrm{ind} + \mathrm{null})(c) = k\,\mathrm{ind}(\sigma) + k \,\mu + n\, . \end{equation}

Note that the homogeneous space $K/K_c$ is diffeomorphic to the orbit $K . c\subseteq \Omega_o M$ of the closed geodesic $c$ under the induced action of $K$ on the based loop space. Moreover, one can also identify $K_c$ with the subgroup

\begin{equation*} \{k\in K\,|\, \mathrm{Ad}_k(\dot{c}(0)) = \dot{c}(0)\} \subseteq K \, , \end{equation*}

see [Reference Ziller35, p. 14]. The group $K$ acts by linear isometries on the space $\mathfrak{m}$, hence the orbit of $\dot{c}(0)$ is diffeomorphic to the space $K/K_c$ which is embedded in the hypersphere

\begin{equation*} S_{c} := \{X\in \mathfrak{m}\,|\, |X| = |\dot{c}(0)|\} \cong \mathbb{S}^{n-1}\, . \end{equation*}

Recall that the dimension of $K/K_c$ is $\mu$. As usual we can assume that $\dot{c}(0)\in \mathfrak{a}$.

If $M$ is an $n$-dimensional symmetric space of rank $1$, then the space $K/K_c$ is the whole sphere $S_c$, since the compact rank one symmetric spaces are precisely the symmetric spaces where all geodesics are closed and of common prime length. Thus $\mu = n-1$ in this case. If $M$ is $n$-dimensional and of rank $r\geq 2$, then the orbit of $K$ cannot be the whole hypersphere $S_c\subseteq \mathfrak{m}$, which one can see as follows. If the orbit of $K$ was the whole $S_c$ then in particular every point in $S_c\cap \mathfrak{a}$ lies in the orbit $K . \dot{c}(0)$. But if $H\in\mathfrak{a}$ is in the $K$-orbit of $\dot{c}(0)$, i.e. there is a $k\in K$ with $H = \mathrm{Ad}_k(\dot{c}(0))$, then there is an element of the Weyl group $s\in W(G,K)$ with $H = s (\dot{c}(0))$, see [Reference Helgason14, Proposition VII.2.2]. But there are only finitely many elements of the Weyl group, so for rank $r\geq 2$ this yields a contradiction to the assumption that the $K$-orbit of $\dot{c}(0)$ is all of $S_c$. Consequently, $K/K_c$ is an embedded closed submanifold of $\mathbb{S}^{n-1}$, which is not the whole sphere and therefore must have positive codimension. We have thus shown the following.

Recall from above that the number $\mu$ shows up in the formulae for index and index plus nullity of the iterates of $c$. In Remark 6.10 we will relate the growth of the index of iteration of closed geodesics in compact symmetric spaces to the results of the next sections concerning the string topology operations on compact symmetric spaces.

5. The Chas-Sullivan product on symmetric spaces

Let $M= G/K$ be a compact symmetric space. As we have seen in the previous section, $M$ is a homogeneous space of the connected component of its isometry group $G$ with $K$ being the stabilizer subgroup of a point. In this section we are going to show that each critical submanifold $\Sigma\subseteq \Lambda M$ consisting of prime closed geodesics of $M$ gives rise to a non-nilpotent element in the Chas-Sullivan algebra of $\Lambda M$. Throughout the latter part of this section we only consider homology with $\mathbb{Z}_2$-coefficients. At the end of this section we shall discuss coefficients in an arbitrary unital commutative ring.

Let $\gamma$ be a closed geodesic in $M$. Let $o = \pi(e)$ be the image of the neutral element $e\in G$ under the canonical projection $\pi \colon G\to M = G/K$. We choose $o$ as the basepoint of $M$. Recall from Section 4 that there is an element $g\in G$ such that $c = g.\gamma \in \Lambda M$ satisfies $c(0) = o$. Since $G$ acts by isometries on $M$, the loop $c$ is also a closed geodesic. Therefore the critical manifold of $c$ in $\Lambda M$ with respect to the energy functional is diffeomorphic to $\Sigma_c = G/K_c$ where $K_c$ is the isotropy group

\begin{equation*} K_c := \{k\in K\,|\, k.c(t) = c(t)\,\,\text{for all}\,\,t\in [0,1]\} \subseteq K\,. \end{equation*}

The projection $\Sigma_c = G/K_c \to G/K$ is a fibre bundle projection

\begin{equation*}K/K_c \hookrightarrow \Sigma_c \xrightarrow[]{\pi} M\end{equation*}

with homogeneous fibre $K/K_c $.

We shall now describe Ziller’s explicit cycles, defined as completing manifolds for the critical manifolds $\Sigma_c$ in [Reference Ziller35, Section 3]. Fix a closed geodesic $c\in \Lambda M$ with $c(0) = o$. Again from Section 4 we recall the following: If $q\in M$ is a point we denote its stabilizer under the action of $K$ by $K_q$, i.e.

\begin{equation*} K_q = \{k\in K\,|\, k.q = q\}\, . \end{equation*}

As Ziller [Reference Ziller35] and Bott and Samelson [Reference Bott and Samelson2] argue, there are finitely many times $0 \lt t_1 \lt \ldots \lt t_l \lt 1$ such that $\mathrm{dim}(K_{c(t_i)}) \gt \mathrm{dim}(K_c)$ for $i\in\{1,\ldots,l\}$. The points $c(t_i)$, $i\in\{1,\ldots,l\}$ are precisely the conjugate points along $c$ and the multiplicity of such a conjugate point is equal to $\mathrm{dim}(K_{c(t_i)}) - \mathrm{dim}(K_c)$. Moreover, the Morse index of the closed geodesic $c$ can be expressed as

\begin{equation*} \mathrm{ind}(c) = \sum_{i= 1}^l \big( \mathrm{dim}(K_{c(t_i)}) - \mathrm{dim}(K_c) \big) \, . \end{equation*}

If $c$ has multiplicity $m$, i.e. there is an underlying prime closed geodesic $\sigma$ with $c=\sigma^m$, then it holds that $c(\tfrac{j}{m}) = o$ for $j\in\{1,\ldots,m-1\}$ and the corresponding stabilizer is the whole group $K$. As before, define $\mu :=\mathrm{dim}(K) - \mathrm{dim}(K_c)$. Recall from Section 4 that the index of the iterates of $\sigma$ equals

\begin{equation*} \mathrm{ind}(c)= \mathrm{ind}(\sigma^m) = m \mathrm{ind}(\sigma) + (m-1)\mu \,. \end{equation*}

Moreover, we have

\begin{equation*} \mathrm{dim}(\Sigma_c) = n + \mu\,, \end{equation*}

where $n$ is the dimension of $M$. We consider the product of groups

\begin{equation*} W_c := G\times K_{c(t_1)}\times K_{c(t_2)} \times \ldots \times K_{c(t_l)}\, . \end{equation*}

There is a right-action of the $(l+1)$-fold product of $K_c$ on $W_c$ given by

(5.1)\begin{equation} \begin{aligned} \chi : W_c \times K_c^{l+1} &\to W_c \\ ( (g_0,k_1,\ldots,k_l), (h_0,\ldots,h_l)) &\mapsto (g_0 h_0,h_0^{-1} k_1 h_1,\ldots, h_{l-1}^{-1} k_l h_l)\,. \end{aligned} \end{equation}

This is a proper and free action and we denote the quotient by

\begin{equation*} \Gamma_c := W_c/ (K_c^{l+1}) \, . \end{equation*}

Note that there is a submersion

(5.2)\begin{equation} p\colon \Gamma_c \to \Sigma_c\cong G/K_c \,, \quad p([(g_0,k_1,\ldots,k_l)]) = [g_0]\, . \end{equation}

Furthermore there is an embedding $\Sigma_c\to \Gamma_c$ defined by

\begin{equation*} s\colon \Sigma_c\to\Gamma_c\,, \quad s([g]) = [(g,e,\ldots,e)] \end{equation*}

and one sees that $p\circ s = \mathrm{id}_{\Sigma_c}$.

We now describe an embedding $ \Gamma_c \hookrightarrow \Lambda M$. Define $\widetilde{f}_c\colon W_c \to \Lambda M$ by

\begin{equation*} \widetilde{f}_c(g_0,k_1,\ldots,k_{l})(t) = \begin{cases} g_0. c(t), & 0\leq t\leq t_1 \\ g_0k_1.c(t), & t_1 \leq t \leq t_2 \\ \,\,\,\,\, \vdots & \,\,\,\,\, \vdots \\ g_0k_1\ldots k_{l}. c(t), & t_l \leq t \leq 1 \,\,\,. \end{cases} \end{equation*}

This factors through the action of $K_c^{l+1}$ and therefore defines a map $f_c\colon \Gamma_c\to \Lambda M$. It can be seen directly that this is an embedding and that $f_c\circ s\colon \Sigma_c\to \Lambda M$ embeds $\Sigma_c$ precisely as the critical manifold. Moreover, the only closed geodesics in $f_c(\Gamma_c)$ are in this critical manifold. Hence, we see that $\Gamma_c$ is a completing manifold if coefficients are taken in $\mathbb{Z}_2$. The same is true with $R$-coefficients if $\Sigma_c$ and $\Gamma_c$ are $R$-orientable as follows from Section 3.

We now show that Ziller’s cycles behave well with respect to the concatenation map

\begin{equation*} \mathrm{concat}_s \colon \Lambda\times_M\Lambda \to \Lambda\,, \quad (\gamma_1,\gamma_2)\mapsto \gamma\,, \end{equation*}

where $s\in(0,1)$ and where

\begin{equation*} \gamma(t) := \begin{cases} \gamma_1(\tfrac{t}{s} ) & 0\leq t\leq s \\ \gamma_2(\tfrac{t+s}{1-s}) & s\leq t\leq 1 \,\,\,. \end{cases} \end{equation*}

Let $\sigma\in \Lambda M$ be a prime closed geodesic and suppose that $\gamma_1,\gamma_2\in \Lambda M$ are iterates of $\sigma$ with multiplicities $m_1$ and $m_2$. Just as above one can set up completing manifolds $\Gamma_1$ and $\Gamma_2$ corresponding to the critical manifolds $\Sigma_1$ and $\Sigma_2$ where $\Sigma_i \subseteq \Lambda M$ is the critical manifold containing $\gamma_i$ for $i\in\{1,2\}$. We want to study the relation between $\Gamma_1,\Gamma_2$ and the completing manifold $\Gamma_3$ associated to the concatenation $\gamma_3 = \sigma^{m_1+m_2}$. We set $m_3 = m_1 + m_2$ and note that we have

\begin{equation*} \Sigma_3 \cong \Sigma_1 \cong \Sigma_2 \cong G/K_{\sigma}\, . \end{equation*}

Recall that we have submersions $p_i\colon \Gamma_i\to \Sigma_i$, $i\in\{1,2,3\}$, and that there is a fiber bundle $\pi\colon G/K_{\sigma}\to M$. This gives submersions $\pi\circ p_i\colon \Gamma_i\to M$ and we can look at the fiber product

\begin{equation*} \Gamma_1\times_M \Gamma_2 \subseteq \Gamma_1\times \Gamma_2 \end{equation*}

with respect to $\pi\circ p_1$ and $\pi\circ p_2$. Clearly, $\Gamma_1\times_M\Gamma_2$ is a submanifold of $\Gamma_1\times\Gamma_2$ of codimension $n$ and it has a canonical map

\begin{equation*} p_1\times p_2\colon \Gamma_1\times_M \Gamma_2 \to \Sigma_1\times_M \Sigma_2, \quad (p_1\times p_2)(X,Y) = (p_1(X),p_2(Y))\, . \end{equation*}

Moreover, we see that

(5.3)\begin{equation} \Gamma_1\times_M\Gamma_2 = \{([(g_0,k_1,\ldots,k_{l_1})],[(\widetilde{g}_0,\widetilde{k}_1,\ldots,\widetilde{k}_{l_2})])\in \Gamma_1\times\Gamma_2\,|\, g_0 K = \widetilde{g}_0 K\} \,. \end{equation}

Recall the construction of the group $W_c$ above for a closed geodesic $c$. If we define $H$ by $ W_{\sigma} = G\times H$ where $\sigma$ is the underlying prime closed geodesic of $\gamma_1$ and $\gamma_2$ as before, then we see that

\begin{equation*} W_{\gamma_i} = G\times H \times (K\times H)^{m_i-1}\,,\quad i\in\{1,2,3\}\, . \end{equation*}

Define a map $\Phi\colon \Gamma_1\times_M \Gamma_2\to \Gamma_3$ by

\begin{equation*} \begin{aligned} \Phi([( g_0,k_1,\ldots,k_{l_1})],&[(\widetilde{g}_0,\widetilde{k}_1,\ldots,\widetilde{k}_{l_2})]) \\ &= [(g_0,k_1,\ldots,k_{l_1} , (g_0 k_1\ldots k_{l_1})^{-1} \widetilde{g}_0, \widetilde{k}_1,\ldots,\widetilde{k}_{l_2})] . \end{aligned}\end{equation*}

Note that the map $\Phi$ is smooth and indeed well-defined since $g_0^{-1}\widetilde{g}_0\in K$ by equation (5.3). Set $\tau = \tfrac{m_1}{m_1+m_2}$ and define $\phi\colon \Sigma_1\times_M\Sigma_2\to \Sigma_3$ by

\begin{equation*} \phi([g],[\widetilde{g}]) = [g] , \end{equation*}

where we identify $\Sigma_i\cong G/K_{\sigma}$ as usual.

Lemma 5.1. The map $\Phi$ is a diffeomorphism. Moreover, it is compatible with the completing manifold structure in the sense that the diagrams

and

commute.

Proof. An explicit inverse for $\Phi$ is given by $\Psi\colon \Gamma_3\to \Gamma_1\times_M\Gamma_2$,

\begin{equation*} \Psi([(g_0,k_1,\ldots,k_{l_3})]) = ([(g_0,k_1,\ldots,k_{l_1})],[((g_0 k_1 \ldots k_{l_1}) k_{l_1 +1} , k_{l_1 + 2},\ldots,k_{l_3}) ]) \, . \end{equation*}

Again, one checks that this is well-defined and smooth and is in fact an inverse to $\Phi$. The commutativity of the diagrams can then be checked by unwinding the definitions.

The property of the completing manifolds described in the following corollary is analogous to the property of the completing manifolds constructed by Oancea for spheres and complex projective spaces in [Reference Oancea28, Section 6].

Corollary 5.2. Let $\sigma$ be a prime closed geodesic of $M$. Let $\Gamma$ denote the completing manifold of the prime closed geodesics isometric to $\sigma$ and let $\Gamma^m$ denote the completing manifold corresponding to $\sigma^m$, for $m\in\mathbb{N}$. We have

\begin{equation*} \Gamma^m\cong\underbrace{\Gamma\times_M\Gamma\times_M\dots\times_M\Gamma}_{m \text{times}}\,. \end{equation*}

In the following we use the notation $(Y,\sim A) := (Y, Y\smallsetminus A)$ for a pair of topological spaces $(Y,A)$ and $\Lambda:=\Lambda M$. Recall from Section 2 that for two homology classes $X\in \mathrm{H}_i(\Lambda M)$ and $Y\in \mathrm{H}_j(\Lambda M)$ their Chas-Sullivan product $X\wedge Y$ is defined by the composition

where $U_{CS}$ is a tubular neighbourhood for the figure eight space $\Lambda \times_M\Lambda\subset\Lambda \times\Lambda$ and the isomorphism in the bottom line is a Thom isomorphism. Consider the following commutative diagram

where the arrows on the left hand side are given by restricting the ones on the right hand side of the diagram. Since $\mathrm{ev}_0\times \mathrm{ev}_0$, $p_1\times p_2$ and $\pi\times\pi$ are submersions it follows by transversality that

\begin{align*} (\mathrm{ev}_0\times \mathrm{ev}_0)^{-1}\left(\Delta(M)\right)&=\Lambda\times_M \Lambda\hookrightarrow\Lambda\times \Lambda\,\\ (f_1\times f_2)^{-1}(\Lambda\times_M \Lambda)=\left((\pi\times\pi)\circ(p_1\times p_2)\right)^{-1}\left(\Delta(M)\right)&=\Gamma_1\times_M \Gamma_2\hookrightarrow\Gamma_1\times \Gamma_2\,\\ (\pi\times \pi)^{-1}\left(\Delta(M)\right)&=\Sigma_1\times_M \Sigma_2\hookrightarrow\Sigma_1\times \Sigma_2\,. \end{align*}

are inclusions of smooth submanifolds of codimension $n$. Let $N_{\Sigma_1\times_M \Sigma_2}$, $N_{\Gamma_1\times_M \Gamma_2}$, $N_{\Lambda\times_M \Lambda}$ denote the corresponding normal bundles. If $N_M$ denotes the normal bundle of the diagonal inclusion $\Delta\colon M\to M\times M$ it also follows that

\begin{align*} (\mathrm{ev}_0\times \mathrm{ev}_0)^*(N_M)&\cong N_{\Lambda\times_M \Lambda}\,\\ (f_1\times f_2)^*(N_{\Lambda\times_M \Lambda})\cong(f_1\times f_2)^*\circ(\mathrm{ev}_0\times \mathrm{ev}_0)^*(N_M) &\\ \cong (p_1\times p_2)^*\circ (\pi\times\pi)^*(N_M) & \cong N_{\Gamma_1\times_M \Gamma_2}\,\\ (\pi\times\pi)^*(N_M)&\cong N_{\Sigma_1\times_M \Sigma_2}\,. \end{align*}

holds. Furthermore, it holds that if $U_M$ is a tubular neighbourhood of $\Delta\colon M\to M\times M$, then

\begin{align*} (\mathrm{ev}_0\times \mathrm{ev}_0)^{-1}\left(U_M\right)=U_{CS}&\text{ is a tubular neighbourhood of }\Lambda\times_M \Lambda\hookrightarrow\Lambda\times \Lambda\,,\\ (f_1\times f_2)^{-1}(U_{CS}) =: U&\text{ is a tubular neighbourhood of }\Gamma_1\times_M \Gamma_2\hookrightarrow\Gamma_1\times \Gamma_2\,. \end{align*}

It then also follows that the diagram

is commutative, where the vertical compositions are the Thom isomorphisms. This can for instance be seen by using a Riemannian submersion metric on $\Gamma_1\times\Gamma_2\to M\times M$. Using Lemma 5.1 this commutative diagram inserts into the following larger commutative diagram:

Here the vertical composition on the right is the Chas-Sullivan product $\wedge$ up to sign.

From now on we only consider homology with $\mathbb{Z}_2$-coefficients.

Proof. We know from Section 3 that the orientation class $[\Sigma_c]$ of a completing manifold of a closed geodesic $c$ injects via $(f_c)_*$ into $\mathrm{H}_*(\Lambda)$. More precisely, one sees that $[\Gamma_c]= (p_c)_!\left([\Sigma_c]\right)$, where $p_c\colon \Gamma_c\to\Sigma_c$ is the projection map from above, see equation (5.2), and consequently

\begin{equation*} (f_c)_*[\Gamma_c] = (f_c)_*\circ (p_c)_! [\Sigma_c] \end{equation*}

is non-zero. We have to show that composition on the left of the above diagram maps $[\Gamma_1]\otimes[\Gamma_2]$ to $[\Gamma_3]$. This can be seen as follows. By the Künneth theorem we have that $[\Gamma_1]\times[\Gamma_2]$ equals $[\Gamma_1\times\Gamma_2]$. The Thom class of $\Gamma_1\times_M\Gamma_2\subset\Gamma_1\times\Gamma_2$ is the Poincaré dual of $[\Gamma_1\times_M\Gamma_2]$ in the homology of $(\bar{U},\partial\bar{U})$ where $\bar{U}$ is the closure of the tubular neighborhood $U$, see [Reference Bredon3, Section VI.11]. Consequently, it follows that capping with the Thom class maps $[\Gamma_1\times\Gamma_2]$ to $[\Gamma_1\times_M\Gamma_2]$. Since $\Phi$ is a diffeomorphism the orientation class of $\Gamma_1\times_M \Gamma_2$ is mapped to the orientation class of $\Gamma_3$.

Corollary 5.4. Let $\sigma$ be a prime closed geodesic, $\Sigma$ the corresponding critical submanifold and $p_\sigma:\Gamma\to\Sigma$ the projection of the completing manifold onto $\Sigma$. Then the class

\begin{equation*} \Theta_\sigma:= (f_\sigma)_*\circ (p_\sigma)_!\left([\Sigma]\right)=(f_\sigma)_*\left([\Gamma]\right)\,. \end{equation*}

is non-nilpotent in the Chas-Sullivan algebra of $\Lambda$.

We now use that $\Sigma_1$ and $\Sigma_2$ are diffeomorphic. Hence, we just write $\Sigma$ to denote either of them. There are even more non-zero Chas-Sullivan products stemming from non-zero products in the intersection algebra of $\Sigma\cong G/K_\sigma$:

Theorem 5.5. If two classes $a,b\in \mathrm{H}_*(\Sigma)$ have nonzero intersection product, then

\begin{equation*} (f_1)_*\circ(p_1)_!(a)\wedge(f_2)_*\circ(p_2)_!(b)\neq 0\,. \end{equation*}

Proof. Consider the inclusions

whose composition gives the diagonal embedding $\Delta_{\Sigma}\colon\Sigma\to\Sigma\times\Sigma$. The map $\Delta'$ is the diagonal section of the map $\phi$ given in Lemma 5.1. The intersection product $a\bullet b$ of $a,b$ is given by

\begin{align*} a\bullet b&= \mathrm{PD}_{\Sigma}\circ(\Delta_{\Sigma})^* \circ\mathrm{PD}_{\Sigma\times\Sigma}^{-1}(a\times b)= \mathrm{PD}_{\Sigma}\circ(\Delta')^* \circ J^* \circ\mathrm{PD}_{\Sigma\times\Sigma}^{-1}(a\times b)\\ &=(\Delta')_!\circ J_!(a\times b)\,. \end{align*}

Let $\lambda_1:= \mathrm{ind}(\sigma^{m_1})$ and $\lambda_2:= \mathrm{ind}(\sigma^{m_2})$ denote the Morse indices of the iterates $\gamma_1=\sigma^{m_1}$ and $\gamma_2=\sigma^{m_2}$ of $\sigma$. The intersection product $a\bullet b$ coincides with left most vertical composition in the following extension of the diagram above, compare Appendix B of [Reference Hingston and Wahl18]:

Apart from the square at the bottom left corner, the above diagram is commutative, since the composition in the centre of the diagram, $\mathrm{H}_{*}(\Gamma_1\times\Gamma_2)\to\mathrm{H}_{*-n}(\Gamma_1\times_M\Gamma_2)$, is the Gysin map of the inclusion. The square marked with $\dagger$ commutes only on the image of the injection

\begin{equation*}\phi_!\colon \mathrm{H}_{*}(\Sigma)\to \mathrm{H}_{*+\mu}(\Sigma\times_M \Sigma)\,,\end{equation*}

i.e. if $y\in \mathrm{H}_{*+\mu}(\Sigma\times_M \Sigma)$ is an element of the image of $\phi_!$, then

\begin{equation*}\Phi_*\circ(p_1\times p_2)_!(y)=(p_3)_!\circ(\Delta')_!(y) \, .\end{equation*}

This follows from the second commutative diagram of Lemma 5.1 and from $\Psi_!=\Phi_*$, where $\Psi$ denotes the inverse of $\Phi$ given in the proof of Lemma 5.1. Since $p_i$ has the section $s_i$ it follows that $(p_i)_!\colon \mathrm{H}_*(\Sigma)\to \mathrm{H}_{*+\lambda_i}(\Gamma_i)$ is injective. Using the sections $s_1\times s_2$ and $(s_1\times s_2)_{|_{\Sigma\times_M \Sigma}}$ the same is true for the maps

\begin{equation*}(p_1\times p_2)_!\colon \mathrm{H}_*(\Sigma\times \Sigma)\to \mathrm{H}_*(\Gamma_1\times\Gamma_2) \end{equation*}

and

\begin{equation*} (p_1\times p_2)_!\colon \mathrm{H}_*(\Sigma\times_M \Sigma)\to \mathrm{H}_*(\Gamma_1\times_M\Gamma_2) \, . \end{equation*}

It follows that all the horizontal compositions in the above diagram are injective, see Remark 3.4. We have that

\begin{equation*} \mathrm{H}_*(\Sigma\times_M\Sigma)\cong\mathrm{ker}\left((\Delta')_!\right)\oplus\mathrm{im}(\phi_!)\cong \mathrm{ker}\left((\Delta')_!\right)\oplus \mathrm{H}_{*-\mu}(\Sigma)\,. \end{equation*}

Let $x+y=J_!(a\times b)$ denote the corresponding splitting of the image of $a\otimes b$ inside $\mathrm{H}_*(\Sigma\times_M\Sigma)$, i.e. $x\in\mathrm{ker}\left((\Delta')_!\right)$ and $y\in\mathrm{im}(\phi_!)$. Then we have

\begin{align*} (f_3)_*\circ(p_3)_!\circ(\Delta')_!(y)&= (f_3)_*\circ(p_3)_!\circ(\Delta')_!(x+y)\\ & = (f_3)_*\circ(p_3)_!\circ(\Delta')_!\left(J_!(a\times b)\right)\\ &= (f_3)_*\circ(p_3)_!(a\bullet b)\neq0\,, \end{align*}

which is nonzero since $(f_3)_*\circ(p_3)_!$ is injective and $a\bullet b\neq 0$ by assumption. On the other hand, by the commutativity of the rest of the diagram we have

\begin{align*} (f_1)_*\circ(p_1)_!(a)\wedge&(f_2)_*\circ(p_2)_!(b)=\mathrm{concat}_*\circ(f_1\times f_2)_*\circ(p_1\times p_2)_!(x+y)\\ &=\mathrm{concat}_*\circ(f_1\times f_2)_*\circ(p_1\times p_2)_!(x)\\ &\quad +\mathrm{concat}_*\circ(f_1\times f_2)_*\circ(p_1\times p_2)_!(y)\\ &=(f_3)_*\circ\Phi_*\circ(p_1\times p_2)_!(x) + (f_3)_*\circ(p_3)_!(a\bullet b)\,. \end{align*}

We now show that this last expression cannot be zero by showing that $(f_3)_*\circ\Phi_*\circ(p_1\times p_2)_!(x)$ cannot be a multiple of $ (f_3)_*\circ(p_3)_!(a\bullet b)$. One checks that the diagram

commutes. It follows that

\begin{equation*}(s_3)_!\circ\Phi_*\circ(p_1\times p_2)_!=(\Delta')_! \, .\end{equation*}

Hence, $\Phi_*\circ(p_1\times p_2)_!(x)$ is an element of the kernel of $(s_3)_!$, since $x\in \mathrm{ker}((\Delta')_!)$. This in turn means that

\begin{equation*} \Phi_*\circ(p_1\times p_2)_!(x) \in \mathrm{H}_*(\Gamma_3\smallsetminus \Sigma) \subseteq \mathrm{H}_*(\Gamma_3) \, .\end{equation*}

The element $(p_3)_!(a\bullet b)$ lies in $\mathrm{H}_*(\Gamma_3,\Gamma_3\smallsetminus\Sigma)\cong \mathrm{H}_{*-\mu}(\Sigma)\cong\mathrm{im}\left((p_3)_!\right)$, i.e. in the other direct summand of $\mathrm{H}_*(\Gamma_3)$, see Section 3. Let $L$ denote the energy of the closed geodesic $\gamma_3$. By Proposition 3.3 we have the direct sum decomposition

\begin{equation*} \mathrm{H}_{*}(\Lambda^{\leq L}) \cong \mathrm{H}_{*}(\Lambda^{ \lt L}) \oplus \mathrm{H}_{*}(\Lambda^{\leq L},X^{ \lt L})\,. \end{equation*}

It follows from Remark 3.4 that under this decomposition $(f_3)_*\circ\Phi_*\circ(p_1\times p_2)_!(x)$ is an element of $\mathrm{H}_*(\Lambda^{ \lt L})$ while $(f_3)_*\circ(p_3)_!(a\bullet b)$ lies in $\mathrm{H}_*(\Lambda^{\leq L},\Lambda^{ \lt L})$. Consequently, the two terms cannot cancel.

A version of the above theorem for a level Chas-Sullivan product in the case of rank one symmetric spaces is Theorem 13.5 in [Reference Goresky and Hingston12]. The following corollary of the theorem is an extension of Proposition 5.3:

Corollary 5.6. For a homology class $a\in \mathrm{H}_*(\Sigma)$ of $\Sigma$ we have

\begin{equation*} (f_1)_*\circ(p_1)_!(a)\wedge(f_2)_*\circ(p_2)_!([\Sigma])=(f_3)_*\circ(p_3)_!(a)\,. \end{equation*}

Proof. We use the notation of the proof of the above theorem. If $J_!(a\times b)=y\in\mathrm{im}(\phi_!)$ then we have

\begin{equation*} (f_1)_*\circ(p_1)_!(a)\wedge(f_2)_*\circ(p_2)_!(b)=(f_3)_*\circ(p_3)_!(a\bullet b)\,. \end{equation*}

This is the case when $b = [\Sigma]$: the intersection product $a\bullet[\Sigma]$ of $[\Sigma]$ with any other class $a$ is given by $(\Delta')_!\circ J_!(a\times[\Sigma])$ and equals $a$. Since the projection onto the first factor $pr_1\colon\Sigma\times\Sigma\to\Sigma$ satisfies $\phi=pr_1\circ J$ and $(pr_1)_!(a)=a\times[\Sigma]$ we have

\begin{equation*} \phi_!(a)=J_!\circ (pr_1)_!(a)=J_!(a\times[\Sigma])\,. \end{equation*}

As the square marked with $\dagger$ in the above diagram commutes on the image on $\phi_!$, the statement of the corollary follows.

The case $a=[\Sigma]$ gives an alternative proof for Proposition 5.3 and Corollary 5.4. Namely, as we have $[\Sigma]\bullet[\Sigma]=[\Sigma]$ it follows from Theorem 5.5 that the class

\begin{equation*} \Theta_\sigma= (f_\sigma)_*\circ (p_\sigma)_!\left([\Sigma]\right)\,, \end{equation*}

which is associated to a prime closed geodesics $\sigma$, satisfies $\Theta_\sigma^{\wedge n}\neq0$ for all $n\in\mathbb{N}$. Here the superscript $\wedge n$ denotes the $n$-fold Chas-Sullivan product.

Combining this with Proposition 4.5 from Section 4, which states that for symmetric spaces of higher rank there are infinitely many geometrically distinct prime critical submanifolds, we can summarize the findings of this section in the following theorem.

Theorem 5.7. Let $M$ be a compact symmetric space and take homology with $\mathbb{Z}_2$-coefficients. Every critical manifold of prime closed geodesics determines a non-nilpotent element in the Chas-Sullivan algebra of $\Lambda M$. If the rank of $M$ is two or larger, there are infinitely many such elements. Such elements also exist if $M$ is only homotopy equivalent to a compact symmetric space, as the Chas-Sullivan product is a homotopy invariant.

Moreover, if $\sigma$ denotes a prime closed geodesic of $M$, then multiplication with $\Theta_\sigma$ gives an isomorphism from the loop space homology generated by $\sigma^m$ to the homology generated by $\sigma^{m+1}$ for every $m\in\mathbb{N}$. More precisely, let $m\in\mathbb{N}$ and let

\begin{equation*}U_{\sigma^m} = \mathrm{im}\left((f_{\sigma^m})_*\circ (p_{\sigma^m})_! \colon \mathrm{H}_*(\Sigma) \hookrightarrow \mathrm{H}_{*+ \mathrm{ind}(\sigma^m)}(\Lambda M) \right)\end{equation*}

be the subspace of $\mathrm{H}_*(\Lambda M)$ induced by the critical manifold $\Sigma_{\sigma^m}$ that contains $\sigma^m$. Then the map

\begin{equation*} \wedge \, \Theta_\sigma \colon U_{\sigma^m} \to U_{\sigma^{m+1}} \,,\,h\mapsto h\wedge\Theta_\sigma\,, \end{equation*}

is an isomorphism for every $m\in\mathbb{N}$.

6. Triviality of the based coproduct for symmetric spaces of higher rank

Using Bott’s $K$-cycles in the based loop space of a compact symmetric space we will show in this section that the based coproduct is trivial for symmetric spaces of higher rank. We will then study the implications of this result for the string topology coproduct on the free loop space. We start by defining Bott’s $K$-cycles in symmetric spaces following [Reference Bott and Samelson2]. See also [Reference Araki1] for more properties of these manifolds.

Let $M = G/K$ be a compact simply connected symmetric space. We adopt all constructions and notations from Section 4. Consider a pair $(\alpha,n)$ where $\alpha\in \Delta$ is a root and $n\in \mathbb{Z}$. As seen in Section 4 this defines an affine hyperplane

\begin{equation*} \{H\in \mathfrak{a}\,|\, \alpha(H) = n \}\subseteq \mathfrak{a} \, , \end{equation*}

the singular plane $(\alpha,n)$. Denote the stabilizer of the maximal torus $T\subseteq M$ under the action of $K$ on $M$ by $K_0$. If $p=(\alpha,n)$ is a singular plane, then let $\overline{p} = \mathrm{Exp}(p)\subseteq T$ and set

\begin{equation*} K_p := \{k\in K\,|\, k.q = q \,\, \text{for all} \,\, q\in \overline{p} \} \, . \end{equation*}

It is clear that both $K_p$ and $K_0$ are closed subgroups of $K$ and that $K_0\subseteq K_p$. Furthermore one can show that

\begin{equation*} \mathrm{dim}(K_p) \gt \mathrm{dim}(K_0)\, , \end{equation*}

for each singular plane $p$, see [Reference Araki1, p. 91].

Let $P = (p_1,\ldots,p_m)$ be an ordered family of singular planes. We set

\begin{equation*} W(P) := K_{p_1}\times K_{p_2} \times \ldots \times K_{p_m}\, . \end{equation*}

There is a right-action of the $m$-fold product of $K_0$ on $W(P)$, that is given by

\begin{eqnarray*} \chi : W(P) \times K_0^m &\to& W(P) \\ ( (k_1,\ldots,k_m), (h_1,\ldots,h_m)) &\mapsto& (k_1 h_1,h_1^{-1} k_2 h_2,\ldots, h_{m-1}^{-1} k_m h_m)\, . \end{eqnarray*}

This is clearly a proper and free action and consequently the quotient

\begin{equation*} \Gamma_P = W(P)/ (K_0^m) \end{equation*}

is a compact smooth manifold. Note the similarity of the construction of these manifolds with the construction of the completing manifolds in Section 5.

We now define an embedding of $\Gamma_P$ into the based loop space $\Omega_o M$. Let $c^P = (c_{0}^P,c_{1}^P,\ldots,c_{m}^P)$ be an ordered family of polygons in $\mathfrak{a}$ such that the following properties are satisfied:

• • The polygon $c_{0}^P$ starts at the origin $0\in\mathfrak{a}$.

• • For $i=1,\ldots,m$ the endpoint of the polygon $c_{i-1}^P$ is the start point of the polygon $c_{i}^P$ and lies on the singular plane $p_i$.

• • The polygon $c_{m}^P$ ends at $0\in\mathfrak{a}$.

For $i=0,1,\ldots,m$, we parametrize the polygon $c_{i}^P$ on the interval $[\frac{i}{m+1},\frac{i+1}{m+1}]$. Define a map $\widetilde{f}_P : W(P) \to \Omega_o M$ by setting

\begin{equation*} \widetilde{f}_P(k_1,k_2,\ldots,k_m) := \mathrm{concat}\Big[ \mathrm{Exp}(c_{0}^P), k_1\mathrm{Exp}(c_{1}^P), \ldots,k_1\ldots k_m\mathrm{Exp}(c_{m}^P) \Big]\, , \end{equation*}

where $\mathrm{concat}$ is the obvious concatenation map of multiple paths. By the particular choice of the polygons, the map $\widetilde{f}_P$ is well-defined and continuous. The map $\widetilde{f}_P$ is invariant under the right-action $\chi$ of $K_0^m$. Consequently it induces a map

\begin{equation*} f_P : \Gamma_P \to \Omega_o M \end{equation*}

and one can check that this is indeed an embedding. Note that the homotopy type of $f_P$ does not depend on the specific choice of polygons $c^P$, see the discussion before Definition 5.4 in [Reference Stegemeyer31].

Let $R$ be a commutative ring with unit. If the manifold $\Gamma_P$ is orientable with respect to $R$ and if $[\Gamma_P]\in \mathrm{H}_{*}(\Gamma_P;R)$ denotes a corresponding fundamental class then we obtain a homology class

\begin{equation*} P_* := (f_P)_*[\Gamma_P] \in \mathrm{H}_{*}(\Omega_o M; R)\, . \end{equation*}

If each $\Gamma_P$ is orientable with respect to $R$, Bott and Samelson show that the set

\begin{equation*} \{P_*\in \mathrm{H}_{*}(\Omega_o M)\,|\, P \,\,\text{is an ordered family of singular planes}\} \end{equation*}

generates the homology $\mathrm{H}_{*}(\Omega_o M)$, see [Reference Bott and Samelson2]. In particular, for every compact symmetric space we get a set that generates the homology of the based loop space with $\mathbb{Z}_2$-coefficients. We will comment on the orientability assumption later on. Note that the empty set is also seen as an ordered family of singular planes and the corresponding homology class corresponds to the basepoint $[o]\in \mathrm{H}_0(\Omega M)$.

We will now show that the basepoint intersection multiplicity of a class $P_*$, where $P$ is an ordered family of singular planes, is $1$. As in Section 4 let $\mathcal{F}\subseteq \mathfrak{a}$ be the lattice

\begin{equation*} \mathcal{F} = \{H\in \mathfrak{a}\,|\, \mathrm{Exp}(H) = o\} \, . \end{equation*}

We make the following definition.

If the rank of $M$ is greater or equal than $2$, i.e. $\mathrm{dim}(\mathfrak{a})\geq 2$, then it is clear that for any non-trivial ordered family of singular planes $P$ we can choose a corresponding ordered family of polygons which is lattice-nonintersecting. Recall that the class $P_*\in \mathrm{H}_{*}(\Omega M)$ does not depend on this choice.

Proof. Choose a representing cycle $X$ of $f_*[\Gamma_P]$ and let $\gamma\in \mathrm{im}(X)$. Since the rank of $M$ is greater or equal than $2$, we can define the class $P_*$ via a lattice-nonintersecting family of polygons $(c_0^P,\ldots,c_m^P)$. Assume that there is $s\in(0,1)$ with $\gamma(s) = o$. Then there are $(k_1,\ldots,k_m)\in W(P)$ with

\begin{equation*} k_1\ldots k_i \mathrm{Exp}(c_i^P(s)) = o \end{equation*}

for some $i\in\{1,\ldots,m\}$. But then $\mathrm{Exp}(c^P_i(s)) = o$, so $c_i^P(s)\in \mathcal{F}$, which contradicts our choice of a lattice-nonintersecting family of polygons.

It is now natural to ask whether the string topology coproduct on the free loop space of compact symmetric spaces of higher rank is trivial as well. Although we cannot answer this question, we will show in the following that the triviality of the based coproduct implies the triviality of the coproduct on a certain subspace of the homology of the free loop space. As seen in Section 2 the diagram

commutes. Hence, it is clear that $\vee$ is trivial on the image of $i_*\colon \mathrm{H}_{*}(\Omega M,o)\to \mathrm{H}_{*}(\Lambda ,M)$ if $\vee_{\Omega}$ is already trivial. Moreover, note that any action $\psi\colon H\times M\to M$ of a compact Lie group $H$ induces an action $\Psi\colon H\times \Lambda M\to \Lambda M$ on the free loop space, which preserves the constant loops. In particular, we obtain an induced pairing in homology

\begin{equation*} \Theta\colon \mathrm{H}_i(H)\otimes \mathrm{H}_j(\Lambda M,M) \to \mathrm{H}_{i+j}(\Lambda M,M),\quad \Theta(X\otimes Y) = \Psi_* ( X\times Y), \end{equation*}

where $ X\in\mathrm{H}_i(H), Y\in \mathrm{H}_j(\Lambda M,M)$.

Proposition 6.6. Let $M$ be a compact simply connected symmetric space of rank greater than or equal to $2$. Consider homology with coefficients in an arbitrary commutative unital ring $R$. Assume that $H$ is a compact Lie group acting on $M$ and consider the pairing $\Theta$ as above. Let $\mathcal{K}$ be the image of $\Theta$ when restricted to the subspace

\begin{equation*} \mathrm{H}_{*}(H) \otimes \big(i_* \,(\mathrm{H}_{*}(\Omega M,o))\big) \subseteq \mathrm{H}_{*}(H)\otimes \mathrm{H}_{*}(\Lambda M,M) . \end{equation*}

Then the string topology coproduct is trivial on $\mathcal{K}$. In particular $i_*(\mathrm{H}_*(\Omega M,o))$ is contained in $\mathcal{K}$.

To conclude this section we show that for a compact symmetric space $M$ we can associate a homology class $[W_{\Sigma}]\in\mathrm{H}_*(\Lambda M,M;\mathbb{Z}_2)$, which lies in $i_*(\mathrm{H}_*(\Omega M,o;\mathbb{Z}_2))$, to any critical manifold $\Sigma\subseteq \Lambda M$. This will be done by using a version of Ziller’s completing manifolds on the based loop space. For the rest of this section we consider homology and cohomology with $\mathbb{Z}_2$-coefficients.

In the following let $M =G/K$ be a compact simply connected symmetric space of arbitrary rank. We saw that the critical manifolds of the energy functional in the free loop space are of the form $\Sigma = G/K_c$, where $c$ is a closed geodesic. Correspondingly, the critical manifolds in the based loop space are of the form $\Sigma_{\Omega} = K/K_c$. We can build a completing manifold $\Gamma_{\Omega}$ in the based loop space just as we did in the free loop space. More explicitly, we set

\begin{equation*} \Gamma_{\Omega} = (K\times K_1\times \ldots K_l)/ (K_c)^{l+1} \end{equation*}

where the groups $K_i$ are the stabilizers of the conjugate points along $c$ exactly as we had it in the case of the free loop space, see Section 5. Moreover, note that the action of the group $(K_c)^{l+1}$ on $K\times K_1\ldots\times K_l$ is obtained by restricting the action $\chi$, which was defined in equation (5.1). If $\Gamma$ is the completing manifold associated to $\Sigma$ then we can embed $\Gamma_{\Omega}$ into $\Gamma$. Moreover, the embedding $f\colon \Gamma\to \Lambda M$ restricts to an embedding of $\Gamma_{\Omega}$ into the based loop space. The submersion $p\colon \Gamma\to \Sigma$ restricts to a submersion $p_{\Omega}\colon \Gamma_{\Omega}\to \Sigma_{\Omega}$. These compatabilities are summarized in the following two commutative diagrams.

Moreover, the section $s\colon \Sigma\to \Gamma$ induces a section $s_{\Omega}\colon \Sigma_{\Omega}\to\Gamma_{\Omega}$ by restriction. All these properties can be checked by using the explicit expressions of the respective maps, which were introduced in Section 5.

Lemma 6.8. The following diagram commutes up to sign.

where $i_{\Gamma}$ and $i_{\Omega}$ are the obvious inclusions.

Proof. First, consider the diagram

(6.1)

and recall that $s_!$ is equal to the composition

\begin{equation*} \mathrm{H}_{i}(\Gamma) \to \mathrm{H}_i(\Gamma,\Gamma\smallsetminus \Sigma) \xrightarrow[]{\text{excision}} \mathrm{H}_i(U,U\smallsetminus \Sigma) \xrightarrow[]{\text{Thom}} \mathrm{H}_{i-\lambda}(\Sigma) \end{equation*}

up to sign. Here, $U$ is a tubular neighborhood of $\Sigma$ in $\Gamma$. The codimension of $\Sigma_{\Omega}$ in $\Gamma_{\Omega}$ is the same as the one of $\Sigma$ in $\Gamma$. The Thom class of $(U,U\smallsetminus \Sigma)$ pulls back to the Thom class of $(U_{\Omega},U_{\Omega}\smallsetminus \Sigma_{\Omega})$ where $U_{\Omega}$ is an appropriatley chosen tubular neighborhood of $\Sigma_{\Omega}$ in $\Gamma_{\Omega}$. Consequently, the above diagram commutes. Now, let $X\in \mathrm{H}_*(\Sigma_{\Omega})$, then we have

\begin{equation*} X = (p_{\Omega}\circ s_{\Omega})_! X = (s_{\Omega})_! (p_{\Omega})_! X , \end{equation*}

since $p_{\Omega}\circ s_{\Omega} = \operatorname{id}_{\Sigma_{\Omega}}$. Thus, we get

\begin{equation*} p_! (i_{\Omega})_* X = p_! (i_{\Omega})_* (s_{\Omega})_! (p_{\Omega})_! X = \pm p_! s_! (i_{\Gamma})_* (p_{\Omega})_! X = \pm (i_{\Gamma})_* (p_{\Omega})_! X \end{equation*}

where we used the commutativity of the diagram (11). This completes the proof.

Proof. Let $\Sigma$ be a critical manifold and denote by $\Gamma$ its associated completing manifold, which comes as usual with the maps $p$ and $f$. We define

\begin{equation*}[W_{\Sigma}] = f_*\circ p_! [x_0] \end{equation*}

where $[x_0]\in\mathrm{H}_0(\Sigma)$ is a generator in degree $0$. Since $\Gamma$ is a completing manifold, we have $[W_{\Sigma}]\neq 0$. By Lemma 6.8 we see that the following diagram commutes.

Clearly the generator $[y_0]\in\mathrm{H}_0(\Sigma_{\Omega})$ in degree $0$ is mapped to $[x_0]$ under $(i_{\Omega})_*$. If we follow the class $[y_0]$ around the diagram we see that

\begin{equation*} [W_{\Sigma}] = i_* \, (f_{\Omega})_* \, (p_{\Omega})_! [y_0], \end{equation*}

so in particular $[W_{\Sigma}]\in i_*(\mathrm{H}_*(\Omega))$. By Proposition 6.6 we then have $\vee[W_{\Sigma}] = 0$.