[Paper Review] Product and coproduct in string topology

Nancy Hingston, Nathalie Wahl|arXiv (Cornell University)|Sep 20, 2017

Black Holes and Theoretical Physics4 citations

TL;DR

This paper extends the Goresky-Hingston cohomology product and homology coproduct on the free loop space of a closed Riemannian manifold to integral singular chains, constructing a non-relative, graded associative and commutative product and coproduct. The key result shows that non-vanishing of the k-th iterate of the coproduct implies the existence of loops with at least (k+1)-fold self-intersections in every representative, and this is sharp for spheres and projective spaces.

ABSTRACT

Let M be a closed Riemannian manifold. We extend the product of Goresky-Hingston, on the cohomology of the free loop space of M relative to the constant loops, to a nonrelative product. It is graded associative and commutative, and compatible with the length filtration on the loop space, like the original product. We prove the following new geometric property of the dual homology coproduct: the nonvanishing of the k-th iterate of the coproduct on a homology class ensures the existence of a loop with a (k+1)-fold self-intersection in every representative of the class. For spheres and projective spaces, we show that this is sharp, in the sense that the k-th iterated coproduct vanishes precisely on those classes that have support in the loops with at most k-fold self-intersections. We study the interactions between this cohomology product and the more well-known Chas-Sullivan product. We give explicit integral chain level constructions of these loop products and coproduct, including a new construction of the Chas-Sullivan product, which avoid the technicalities of infinite dimensional tubular neighborhoods and delicate intersections of chains in loop spaces.

Motivation & Objective

• To lift the relative Goresky-Hingston cohomology product and homology coproduct to integral singular chains on the free loop space.
• To define a non-relative, graded associative and commutative cohomology product and homology coproduct compatible with the length filtration.
• To establish a geometric link between the iterated coproduct and the minimal number of self-intersections in loop representatives.
• To provide explicit, geometric chain-level constructions of the Chas-Sullivan product and coproduct without relying on infinite-dimensional tubular neighborhoods or delicate intersection limits.
• To demonstrate that for spheres and projective spaces, the k-th iterated coproduct vanishes precisely on classes supported by loops with at most k-fold self-intersections.

Proposed method

• Construct the Chas-Sullivan product and coproduct using retraction maps derived from a tubular neighborhood of the diagonal in M×M.
• Define the cohomology product via a Thom class cap followed by concatenation of loops after retraction to common basepoints.
• Define the homology coproduct via a sign-corrected cut map applied to loops with self-intersections, after retracting to the diagonal via geodesic sticks.
• Use the evaluation map ΛM → M to split homology and cohomology, enabling extension by zero to define the non-relative operations.
• Lift the relative operations to integral chains via extension by zero on constant loops, ensuring compatibility with the length filtration.
• Prove associativity, graded commutativity, and compatibility with the length filtration using sign-corrected chain-level constructions and diagram chases with cap and cross products.

Experimental results

Research questions

• RQ1 Does the iterated coproduct detect the minimal number of self-intersections in loop representatives?
• RQ2 Can the Chas-Sullivan product and Goresky-Hingston coproduct be constructed at the integral chain level without infinite-dimensional tubular neighborhoods?
• RQ3 Is the non-vanishing of the k-th iterate of the coproduct sufficient to guarantee a representative with a (k+1)-fold self-intersection?
• RQ4 Does the coproduct vanish exactly on classes supported by loops with at most k-fold self-intersections for spheres and projective spaces?
• RQ5 How do the new chain-level products and coproducts interact algebraically, particularly with respect to the Frobenius identity?

Key findings

• The non-vanishing of the k-th iterate of the coproduct on a homology class implies that every representative of the class contains a loop with at least (k+1)-fold self-intersection.
• For spheres and complex projective spaces, the k-th iterated coproduct vanishes if and only if the class is supported by loops with at most k-fold self-intersections, making the invariant sharp.
• The constructed cohomology product b⊛ is graded associative and commutative, and compatible with the length filtration on the loop space.
• The coproduct b∨ satisfies the identity ∧◦b∨=0, indicating a form of involutive duality with the Chas-Sullivan product.
• The paper provides a new, geometric chain-level construction of the Chas-Sullivan product that avoids technicalities of infinite-dimensional tubular neighborhoods and delicate intersection limits.
• The sign corrections in the definitions ensure that the product and coproduct satisfy the expected algebraic properties, including associativity and graded commutativity.