[Paper Review] A Note on the Chas-Sullivan product

François Laudenbach|arXiv (Cornell University)|Mar 16, 2009

Geometric and Algebraic Topology5 references3 citations

TL;DR

This paper presents a finite-dimensional, simplicial approach to the Chas-Sullivan product on the free loop space of a manifold, using transversality and small simplices with local coefficients to define the product at the chain level. The key contribution is a homotopy-invariant, geometrically intuitive construction that extends to non-orientable manifolds and recovers known results via spectral sequences and Thom isomorphisms.

ABSTRACT

We give a finite dimensional approach to the Chas-Sullivan product on the free loop space of a manifold, orientable or not.

Motivation & Objective

• To provide a finite-dimensional, geometrically intuitive construction of the Chas-Sullivan product on the free loop space of a manifold.
• To extend the product to non-orientable manifolds using local coefficients instead of integer coefficients.
• To define the product at the chain level using transversality and small simplices, avoiding infinite-dimensional methods.
• To show that this construction yields a well-defined homology-level product isomorphic to known string topology spectral sequences.
• To recover and reinterpret results from Chataur-Le Borgne and others using this simplicial, transversality-based framework.

Proposed method

• The free loop space $ LM $ is treated as a simplicial set, with $ k $-simplices defined as smooth maps $ riangle^k \times S^1 \to M $, and the evaluation map $ ev_0 $ sends such a simplex to its restriction at $ S^1 $'s basepoint.
• Small simplices are defined as those whose image lies within a single chart of a given atlas, ensuring local control and compatibility with local coefficients.
• Transversality is applied to bi-simplices in $ LM \times LM $, specifically to those mapping to composable loops in $ M $, to define an intersection product at the chain level.
• The intersection of transverse cycles in $ LM \times_M LM $ yields a smooth manifold $ W $, which is triangulated via Whitehead's uniqueness result, yielding a well-defined cycle in $ LM $.
• The Thom isomorphism is used to relate the spectral sequence page $ E^1 $ to the intersection homology of $ M $ and its unit normal bundle $ UM $, enabling a bigraded ring isomorphism.
• The spectral sequence $ E^1 $-page is shown to be isomorphic to $ \mathbb{H}_*(M;\mathbb{Z}_{or}) \oplus \mathbb{H}_*(UM;\mathbb{Z})[T]_{\geq 1} $, with $ d^1 = 0 $, implying $ E^\infty \cong E^1 $.

Experimental results

Research questions

• RQ1 Can the Chas-Sullivan product be defined directly at the chain level using finite-dimensional, simplicial methods rather than infinite-dimensional homotopy theory?
• RQ2 How can the product be extended to non-orientable manifolds without relying on orientable structures?
• RQ3 What is the role of transversality and small simplices in ensuring the product is well-defined and homotopy-invariant?
• RQ4 How does the spectral sequence of the fibration $ \Lambda M \to M $ relate to the intersection ring structure on $ UM $ and $ M $?
• RQ5 Can known results in string topology, such as those by Chataur and Le Borgne, be recovered and reinterpreted via this simplicial, transversality-based framework?

Key findings

• The Chas-Sullivan product is well-defined at the homology level via triangulation of transverse intersections in $ LM \times_M LM $, using Whitehead's uniqueness of triangulations up to isotopy and subdivision.
• The spectral sequence $ E^1 $-page is isomorphic to $ \mathbb{H}_*(M;\mathbb{Z}_{or}) \oplus \mathbb{H}_*(UM;\mathbb{Z})[T]_{\geq 1} $, where $ T $ has bidegree $ (1, \alpha_1 + n - 2) $, and the ring structure is induced by the Thom isomorphism.
• The differential $ d^1 $ vanishes identically because $ \mathbb{H}_*(\Lambda_0) $ is a direct summand, and $ d^1 $ is a derivation, so $ d^1(T) = 0 $, implying $ d^1 = 0 $ everywhere.
• The spectral sequence collapses at $ E^1 $, so $ E^\infty \cong E^1 $, and the final bigraded ring structure matches the initial one.
• The construction is valid for both orientable and non-orientable manifolds, using local coefficients $ \mathbb{Z}_{or} $ to handle orientation issues.
• The method recovers and reinterprets results from Chataur and Le Borgne, showing that the simplicial, transversality-based approach is as efficient as infinite-dimensional methods for concrete geometric computations.