[Paper Review] Loop coproducts in string topology and triviality of higher genus TQFT operations

Hirotaka Tamanoi|ArXiv.org|Jun 8, 2007

Logic, programming, and type systems5 references3 citations

TL;DR

This paper establishes that all string topology operations associated with oriented surfaces of genus at least one vanish identically, due to the triviality of the loop coproduct outside the degree-d homology of contractible loops. The loop coproduct is shown to be nontrivial only on the component of contractible loops and maps exclusively to constant loop classes, with values scaled by the Euler characteristic of the manifold, leading to the vanishing of higher genus TQFT operations.

ABSTRACT

Cohen and Godin constructed positive boundary topological quantum field theory (TQFT) structure on the homology of free loop spaces of oriented closed smooth manifolds by associating a certain operations called string operations to orientable surfaces with parametrized boundaries. We show that all TQFT string operations associated to surfaces of genus at least one vanish identically. This is a simple consequence of properties of the loop coproduct which will be discussed in detail. One interesting property is that the loop coproduct is nontrivial only on the degree $d$ homology group of the connected component of $LM$ consisting of contractible loops, where $d=\dim M$, with values in the degree 0 homology group of constant loops. Thus the loop coproduct behaves in a dramatically simpler way than the loop product.

Motivation & Objective

• To understand the structure and behavior of the loop coproduct in string topology.
• To investigate the implications of the loop coproduct for higher genus topological quantum field theory (TQFT) string operations.
• To determine under what conditions string operations associated with surfaces of genus g ≥ 1 vanish.
• To clarify the role of the Euler characteristic in the vanishing of these operations.
• To establish the precise domain and image of the loop coproduct in homology.

Proposed method

• The loop coproduct is analyzed via its action on homology classes of free loop spaces, particularly focusing on the component of contractible loops.
• The paper uses Frobenius compatibility and coderivation properties of cap products to derive structural constraints on the loop coproduct.
• It applies the homotopy-theoretic definition of the loop product and extends it to coproducts via pants decompositions.
• The key equation is Ψ(a₁⋯aₚ) = χ(M)[c₀]a₁⋯aₗ ⊗ [c₀]aₗ₊₁⋯aₚ, which defines the loop coproduct in terms of the Euler characteristic and constant loop classes.
• Vanishing results are derived by showing that χ(M)[c₀]·x = 0 for x not in degree 0, especially when χ(M) = 0 or in torsion-free settings.
• The proof leverages the triviality of the S¹-action on constant loops and the fact that Δ([c₀]) = 0.

Experimental results

Research questions

• RQ1 Does the loop coproduct vanish on all components of the free loop space except the contractible loop component?
• RQ2 Under what conditions do string operations associated with genus-g surfaces with g ≥ 1 vanish?
• RQ3 How does the Euler characteristic χ(M) influence the behavior of the loop coproduct?
• RQ4 Is the loop coproduct compatible with the BV operator and cap products in a nontrivial way?
• RQ5 What is the failure of commutativity in the diagram involving Δ and Ψ, and does it vanish?

Key findings

• All string operations associated with surfaces of genus g ≥ 1 or with q ≥ 3 outgoing boundaries vanish identically.
• The loop coproduct Ψ is nontrivial only on H_d((LM)_[1]), the degree-d homology of the contractible loop component.
• The image of Ψ lies entirely in H₀((LM)_[1]) ⊗ H₀((LM)_[1]) ≅ ℤ[c₀] ⊗ [c₀], the homology of constant loops.
• For manifolds with vanishing Euler characteristic, such as odd-dimensional manifolds, the loop coproduct is identically zero.
• The identity (Δ×1 + 1×Δ)Ψ(a) = 0 holds for all a ∈ H_*(LM), showing triviality of the coproduct's compatibility with the BV operator.
• When a ∈ H_*(M), the loop coproduct of the loop bracket {a,b} vanishes: Ψ({a,b}) = 0.