[Paper Review] Transversality problems in string topology and de Rham chains

Kei Irie|arXiv (Cornell University)|Apr 1, 2014

Homotopy and Cohomology in Algebraic Topology7 references3 citations

TL;DR

This paper introduces a de Rham chain complex that unifies singular and de Rham homology to resolve transversality issues in string topology. By constructing a chain complex on the loop space using this hybrid framework, the authors define string topology operations without perturbations, ensuring compatibility with the length filtration and enabling a rigorous, geometrically natural formulation of operations in string topology.

ABSTRACT

We propose a new approach to resolve transversality problems in string topology. For this purpose, we introduce a notion of de Rham chain complex, which is a hybrid of the singular chain complex and the usual de Rham complex. Using this machinery, we define a chain complex of the loop space, on which one can define string topology operations without any perturbations. All operations are defined so that they respect the length filtration on the loop space.

Motivation & Objective

• To address persistent transversality problems in string topology that hinder the construction of well-defined chain-level operations.
• To develop a unified homological framework combining singular and de Rham chains for the loop space.
• To define string topology operations intrinsically, without relying on perturbation techniques.
• To ensure all operations respect the natural length filtration on the loop space.
• To provide a geometrically meaningful, cohomologically structured chain model for string topology operations.

Proposed method

• Introduce a new de Rham chain complex that interpolates between singular chains and differential forms.
• Construct a chain complex on the free loop space using this hybrid de Rham chain framework.
• Define string topology operations (e.g., loop product, BV operator) directly on the chain complex without perturbation.
• Ensure all operations are compatible with the length filtration by leveraging the geometric structure of the de Rham chains.
• Use the de Rham chain complex to model the loop space in a way that preserves both topological and differential geometric data.
• Establish that the resulting complex supports a well-defined, filtration-preserving algebraic structure reflecting string topology operations.

Experimental results

Research questions

• RQ1 How can transversality issues in string topology be resolved at the chain level without ad hoc perturbations?
• RQ2 Can a unified chain complex be constructed that combines the strengths of singular and de Rham homology for loop spaces?
• RQ3 Do string topology operations defined via this new complex respect the natural length filtration on the loop space?
• RQ4 Is it possible to define the loop product and BV operator intrinsically using a geometric chain model?
• RQ5 What algebraic structure emerges from the de Rham chain complex of the loop space in the context of string topology?

Key findings

• The de Rham chain complex successfully resolves transversality problems in string topology by providing a geometric, perturbation-free framework.
• String topology operations, including the loop product and BV operator, are defined directly on the chain complex without requiring transversality assumptions.
• All operations are shown to preserve the length filtration on the loop space, ensuring compatibility with geometric and topological structures.
• The construction yields a well-defined chain-level model for string topology that unifies singular and de Rham data.
• The framework enables a natural, intrinsic definition of string topology operations that are compatible with the underlying geometry of the loop space.
• The resulting complex supports a consistent algebraic structure that reflects the known operations in string topology, now at the chain level without perturbation.