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String topology on the space of paths with endpoints in a submanifold

Maximilian Stegemeyer

TL;DR

This work develops a general framework for string topology on the space of paths with endpoints in a submanifold by studying the pullback path space $P^f$. It defines a path product $\wedge$ on $H_\bullet(P^f)$ and a compatible left module structure over the Chas-Sullivan ring, proving invariance under homotopies of $f$ and deriving explicit descriptions in key cases, notably for null-homotopic embeddings where $H_\bullet(P^f)$ often becomes an algebra over the CS ring. The diagonal map case recovers the classical CS product, while fiber-bundle and constant-map examples yield concrete tensor-product models $H_\bullet(N)^{\otimes 2}\otimes H_\bullet(\Omega M)$ and related constructions. Together, these results provide computable tools for path-space string topology and link the path-space algebra to familiar algebraic structures via explicit Thom and Gysin treatments.

Abstract

In this article we consider algebraic structures on the homology of the space of paths in a manifold with endpoints in a submanifold. The Pontryagin-Chas-Sullivan product on the homology of this space had already been investigated by Hingston and Oancea for a particular example. We consider this product as a special case of a more general construction where we consider pullbacks of the path space of a manifold under arbitrary maps. The product on the homology of this space as well as the module structure over the Chas-Sullivan ring are shown to be invariant under homotopies of the respective maps. This in particular implies that the Pontryagin-Chas-Sullivan product as well as the module structure on the space of paths with endpoints in a submanifold are isomorphic for two homotopic embeddings of the submanifold. Moreover, for null-homotopic embeddings of the submanifold this yields nice formulas which we can be used to compute the product and the module structure explicitly. We show that in the case of a null-homotopic embedding the homology of the space of paths with endpoints in a submanifold is even an algebra over the Chas-Sullivan ring.

String topology on the space of paths with endpoints in a submanifold

TL;DR

This work develops a general framework for string topology on the space of paths with endpoints in a submanifold by studying the pullback path space . It defines a path product on and a compatible left module structure over the Chas-Sullivan ring, proving invariance under homotopies of and deriving explicit descriptions in key cases, notably for null-homotopic embeddings where often becomes an algebra over the CS ring. The diagonal map case recovers the classical CS product, while fiber-bundle and constant-map examples yield concrete tensor-product models and related constructions. Together, these results provide computable tools for path-space string topology and link the path-space algebra to familiar algebraic structures via explicit Thom and Gysin treatments.

Abstract

In this article we consider algebraic structures on the homology of the space of paths in a manifold with endpoints in a submanifold. The Pontryagin-Chas-Sullivan product on the homology of this space had already been investigated by Hingston and Oancea for a particular example. We consider this product as a special case of a more general construction where we consider pullbacks of the path space of a manifold under arbitrary maps. The product on the homology of this space as well as the module structure over the Chas-Sullivan ring are shown to be invariant under homotopies of the respective maps. This in particular implies that the Pontryagin-Chas-Sullivan product as well as the module structure on the space of paths with endpoints in a submanifold are isomorphic for two homotopic embeddings of the submanifold. Moreover, for null-homotopic embeddings of the submanifold this yields nice formulas which we can be used to compute the product and the module structure explicitly. We show that in the case of a null-homotopic embedding the homology of the space of paths with endpoints in a submanifold is even an algebra over the Chas-Sullivan ring.
Paper Structure (13 sections, 28 theorems, 238 equations, 1 figure)

This paper contains 13 sections, 28 theorems, 238 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Definition and basic properties of the path product
  3. Examples of the path product
  4. The constant map
  5. Fiber bundles
  6. The diagonal map
  7. Invariance of the path product
  8. Module structure over the Chas-Sullivan ring
  9. The module structure in particular cases
  10. The constant map
  11. Embedding of a submanifold
  12. The identity map
  13. Invariance of the module structure

Key Result

Theorem 1

Let $M$ and $N$ be closed oriented manifolds and assume that $f\colon N\to M$ and $g\colon N\to M$ are smooth maps. Take homology with coefficients in a field $R$ and consider the homology of the spaces $P^f$ and $P^g$. If $f$ and $g$ are homotopic then

Figures (1)

  • Figure 1: The path $\Xi(x_0,\gamma(0))$. The dashed paths labelled (1) and (2) are the geodesic sticks from $\gamma(0)$ to $f(x_0)$ and from $g(x_0)$ to $\xi(x_0,\gamma(0))$, respectively. The dashed path labelled $\eta(x_0)$ is the path defined by the homotopy $H$ between $f(x_0)$ and $g(x_0)$.

Theorems & Definitions (57)

  • Theorem : Theorem \ref{['theorem_invariance_path_product']} and Theorem \ref{['theorem_invariance_module_structure']}
  • Theorem : Proposition \ref{['prop_trivial_map_path_space']} and Proposition \ref{['prop_trivial_map_module']}
  • Theorem : Theorem \ref{['theorem_algebra_over_cs']}
  • Remark 2.1
  • Definition 2.2
  • Remark 2.3
  • Proposition 2.4
  • Theorem 2.5
  • proof
  • Remark 2.6
  • ...and 47 more