[Paper Review] The string topology BV algebra, Hochschild cohomology and the Goldman bracket on surfaces

Dmitry Vaintrob|ArXiv.org|Feb 28, 2007

Algebraic structures and combinatorial models17 references18 citations

TL;DR

This paper establishes a natural isomorphism between the string topology Batalin-Vilkovisky (BV) algebra on the free loop space homology of an aspherical closed oriented manifold and the Hochschild cohomology BV algebra of the group algebra of its fundamental group. Using topological and algebraic constructions, the authors prove that the Chas-Sullivan BV operator on loop homology corresponds to the Connes operator dualized via Poincaré duality, and that the loop product maps to the cup product, fully identifying the BV algebra structures on both sides for hyperbolic surfaces and more general aspherical manifolds.

ABSTRACT

In 1999 Chas and Sullivan discovered that the homology H_*(LX) of the space of free loops on a closed oriented smooth manifold X has a rich algebraic structure called string topology. They proved that H_*(LX) is naturally a Batalin-Vilkovisky (BV) algebra. There are several conjectures connecting the string topology BV algebra with algebraic structures on the Hochschild cohomology of algebras related to the manifold X, but none of them has been verified for manifolds of dimension n>1. In this work we study string topology in the case when X is aspherical (i.e. its homotopy groups π_i(X) vanish for i > 1). In this case the Hochschild cohomology Gerstenhaber algebra HH^*(A) of the group algebra A of the fundamental group of X has a BV structure. Our main result is a theorem establishing a natural isomorphism between the Hochschild cohomology BV algebra HH^*(A) and the string topology BV algebra H_*(LX). In particular, for a closed oriented surface X of hyperbolic type we obtain a complete description of the BV algebra operations on H_*(LX) and HH^*(A) in terms of the Goldman bracket of loops on X. The only manifolds for which the BV algebra structure on H_*(LX) was known before were spheres and complex Stiefel manifolds. Our proof is based on a combination of topological and algebraic constructions allowing us to compute and compare multiplications and BV operators on both H_*(LX) and HH^*(A).

Motivation & Objective

• To resolve the long-standing conjecture that the string topology BV algebra on loop homology is isomorphic to Hochschild cohomology BV algebra for manifolds of dimension >1.
• To provide the first explicit computation of the full BV algebra structure on loop homology for manifolds beyond spheres and complex Stiefel manifolds.
• To establish a precise correspondence between the Chas-Sullivan BV operator and the Connes operator in Hochschild cohomology via Poincaré duality.
• To demonstrate that the loop product and Gerstenhaber bracket on loop homology correspond to the cup product and Lie bracket on Hochschild cohomology.
• To extend the understanding of string topology operations by connecting them algebraically to group algebra cohomology for aspherical manifolds.

Proposed method

• Construct a vector space isomorphism ρ between the loop homology H_*(LX) and the Hochschild homology HH_*(A) of the group algebra A = k[π₁(X)].
• Show that ρ maps the Chas-Sullivan BV operator Δ on H_*(LX) to the Connes operator κ on HH_*(A).
• Define a Poincaré duality isomorphism τ: HH_*(A) → HH^{n-*}(A) using a dual resolution construction.
• Use τ to induce a BV operator λ = τ ∘ κ ∘ τ⁻¹ on Hochschild cohomology HH^{*}(A), proving it is compatible with the Gerstenhaber algebra structure.
• Prove that the composition ξ = τ ∘ ρ: H_*(LX) → HH^{n-*}(A) is an isomorphism of associative algebras, preserving both the product and the BV operator.
• Demonstrate commutativity of diagrams linking the loop product, cup product, and the respective BV structures, confirming full BV algebra isomorphism.

Experimental results

Research questions

• RQ1 Is there a natural isomorphism between the string topology BV algebra on H_*(LX) and the Hochschild cohomology BV algebra HH^{*}(A) for aspherical manifolds?
• RQ2 Does the Chas-Sullivan BV operator Δ on loop homology correspond to the Connes operator κ on Hochschild homology under Poincaré duality?
• RQ3 Can the loop product on H_*(LX) be identified with the cup product on HH^{*}(A) under the same isomorphism?
• RQ4 For closed oriented surfaces of hyperbolic type, can the full BV algebra structure be described explicitly via the Goldman bracket?
• RQ5 Does the isomorphism preserve the Gerstenhaber bracket structure between string topology and Hochschild cohomology?

Key findings

• The paper establishes a natural isomorphism between the string topology BV algebra H_*(LX) and the Hochschild cohomology BV algebra HH^{*}(A) for any aspherical closed oriented manifold X.
• The isomorphism ξ: H_*(LX) → HH^{n-*}(A) preserves both the associative product and the BV operator, confirming that the full BV algebra structures are isomorphic.
• The Chas-Sullivan BV operator Δ on loop homology is mapped to the Connes operator κ on Hochschild homology under the isomorphism ρ.
• The Poincaré duality isomorphism τ: HH_*(A) → HH^{n-*}(A) induces a BV operator λ on cohomology that is compatible with the Gerstenhaber algebra structure.
• For hyperbolic surfaces, the BV algebra operations on both H_*(LX) and HH^{*}(A) are fully described in terms of the Goldman bracket.
• The isomorphism ξ maps the Goldman bracket on loops to the Gerstenhaber bracket on Hochschild cohomology, confirming compatibility of the Lie-type structures.