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On the string topology of symmetric spaces of higher rank

Philippe Kupper, Maximilian Stegemeyer

TL;DR

This work extends string topology to higher-rank compact symmetric spaces by leveraging Bott–Samelson and Ziller cycles to realize the Morse–Bott structure of the energy functional on loop spaces. It proves that the Chas–Sullivan product is highly non-trivial in this setting, yielding infinitely many non-nilpotent classes whose powers reflect iterations of closed geodesics, while the based coproduct is trivial for rank at least two, shaping the behavior of the free loop coproduct. By constructing explicit completing manifolds and Bott’s K-cycles, the paper derives precise algebraic decompositions and shows that coproduct vanishing occurs on large subspaces, including products of symmetric spaces. The results clarify the contrast between rank-one and higher-rank cases and contribute to understanding BV-algebra structures in string topology for symmetric spaces.

Abstract

The homology of the free and the based loop space of a compact globally symmetric space can be studied through explicit cycles. We use cycles constructed by Bott and Samelson and by Ziller to study the string topology coproduct and the Chas-Sullivan product on compact symmetric spaces. We show that the Chas-Sullivan product for compact symmetric spaces is highly non-trivial for any rank and we prove that there are many non-nilpotent classes whose powers correspond to the iteration of closed geodesics. Moreover, we show that the based string topology coproduct is trivial for compact symmetric spaces of higher rank and we study the implications of this result for the string topology coproduct on the free loop space.

On the string topology of symmetric spaces of higher rank

TL;DR

This work extends string topology to higher-rank compact symmetric spaces by leveraging Bott–Samelson and Ziller cycles to realize the Morse–Bott structure of the energy functional on loop spaces. It proves that the Chas–Sullivan product is highly non-trivial in this setting, yielding infinitely many non-nilpotent classes whose powers reflect iterations of closed geodesics, while the based coproduct is trivial for rank at least two, shaping the behavior of the free loop coproduct. By constructing explicit completing manifolds and Bott’s K-cycles, the paper derives precise algebraic decompositions and shows that coproduct vanishing occurs on large subspaces, including products of symmetric spaces. The results clarify the contrast between rank-one and higher-rank cases and contribute to understanding BV-algebra structures in string topology for symmetric spaces.

Abstract

The homology of the free and the based loop space of a compact globally symmetric space can be studied through explicit cycles. We use cycles constructed by Bott and Samelson and by Ziller to study the string topology coproduct and the Chas-Sullivan product on compact symmetric spaces. We show that the Chas-Sullivan product for compact symmetric spaces is highly non-trivial for any rank and we prove that there are many non-nilpotent classes whose powers correspond to the iteration of closed geodesics. Moreover, we show that the based string topology coproduct is trivial for compact symmetric spaces of higher rank and we study the implications of this result for the string topology coproduct on the free loop space.
Paper Structure (7 sections, 26 theorems, 176 equations, 2 figures)

This paper contains 7 sections, 26 theorems, 176 equations, 2 figures.

Table of Contents

  1. Introduction
  2. String topology product and coproduct
  3. Completing manifolds
  4. Geodesics in symmetric spaces
  5. The Chas-Sullivan product on symmetric spaces
  6. Triviality of the based coproduct for symmetric spaces of higher rank
  7. The string topology coproduct on products of symmetric spaces

Key Result

Theorem 1

Let $M$ be a compact symmetric space and take homology with $\mathbb{Z}_2$-coefficients. For every critical manifold $\Sigma_{\gamma}$ the associated class $\Theta =(f_{\gamma})_*[\Gamma_{\gamma}]$ is non-nilpotent in the Chas-Sullivan algebra and the powers of this class correspond to the iteration Moreover, for each $m\in\mathbb{N}$ the multiplication with $\Theta$ gives an isomorphism from the

Figures (2)

  • 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 Sakai3 and Sakai1. 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 $\mathop{\mathrm{Exp}}\nolimits$ 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.
  • Figure 2: Example of the positive Weyl chamber for the case of the maximal abelian subspace shown in Figure \ref{['FigureRootSystem']}. The Weyl chamber is depicted as the shaded area. The lattice points are denoted as bullet points. Note that the closed geodesic $\gamma_H = \mathop{\mathrm{Exp}}\nolimits\circ \sigma_H$ 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 $(\epsilon,0)$.

Theorems & Definitions (57)

  • Theorem : Theorem \ref{['nonnilpotent_CS_classes']}
  • Theorem : Corollary \ref{['cor_triviality_based']} and Proposition \ref{['prop_wc_classes']}
  • Theorem : Theorem \ref{['theorem_intersection_product_completing']}
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • Proposition 2.7: hingston:2017, Theorem 3.10
  • ...and 47 more