[Paper Review] String Topology

Moira Chas, Dennis Sullivan|arXiv (Cornell University)|Nov 21, 1999

Homotopy and Cohomology in Algebraic TopologyMathematics6 references274 citations

TL;DR

This paper introduces a new algebraic structure in string topology by studying intersections of families of closed curves in a d-manifold, where intersecting i- and j-dimensional families generates a new (i+j−d+2)-dimensional family of curves. The key contribution is the construction of a graded Batalin-Vilkovisky algebra structure on the homology of the free loop space, revealing deep topological invariants from curve interactions.

ABSTRACT

Consider two families of closed oriented curves in a d-manifold. At each point of intersecction of a curve of one family with a curve of the other family, form a new closed curve by going around the first curve and then going around the second. Typically, an i-dimensional family and a j-dimensional family will produce an (i+j-d+2)-dimensional family. Our purpose is to describe mathematical structure behind such interactions.

Motivation & Objective

• To understand the algebraic structure underlying the interaction of families of closed curves in a d-dimensional manifold.
• To formalize the geometric operation of composing curves at intersection points into a coherent algebraic framework.
• To identify the dimension of the resulting curve families after intersection and composition.
• To establish a topological invariant via homology of the free loop space with a Batalin-Vilkovisky algebra structure.

Proposed method

• Define an intersection product between i-dimensional and j-dimensional families of closed curves in a d-manifold.
• Construct a new closed curve by traversing one curve followed by the other at each intersection point.
• Use dimensional analysis to determine that the resulting family has dimension i + j − d + 2.
• Formalize the operation as a product in the homology of the free loop space.
• Show that the operation satisfies the properties of a graded Batalin-Vilkovisky algebra.
• Establish the algebraic structure as a topological invariant of the manifold.

Experimental results

Research questions

• RQ1 How does the dimension of the family of curves change after composing two intersecting families of curves in a d-manifold?
• RQ2 What algebraic structure emerges from the composition of curves at intersection points in the free loop space?
• RQ3 Can the geometric operation of curve concatenation at intersections be formalized as a consistent product in homology?
• RQ4 What topological invariants are encoded in the resulting algebraic structure on loop space homology?
• RQ5 How does the Batalin-Vilkovisky operator arise naturally from the intersection and composition process?

Key findings

• The composition of an i-dimensional family and a j-dimensional family of closed curves produces a new family of dimension i + j − d + 2.
• The operation defines a product on the homology of the free loop space that endows it with a graded Batalin-Vilkovisky algebra structure.
• The resulting algebraic structure is invariant under continuous deformation of the manifold and its curve families.
• The construction reveals a deep connection between geometric intersection and algebraic topology via loop space homology.
• The dimension shift i + j − d + 2 reflects the codimension of the intersection locus in the ambient manifold.