[Paper Review] Rational BV-algebra in String Topology

Yves Félix, Jean‐Claude Thomas|ArXiv.org|May 29, 2007

Homotopy and Cohomology in Algebraic Topology16 references4 citations

TL;DR

This paper establishes a rational BV-algebra structure on the Hochschild cohomology $\mathrm{HH}^\ast(C^\ast(M); C^\ast(M))$ for a 1-connected closed manifold $M$, proving it is isomorphic to the shifted homology $H_{\ast+m}(LM)$ as BV-algebras when coefficients are in a field of characteristic zero. The construction uses rational homotopy theory and a Poincaré duality model to lift the Chas-Sullivan BV structure from string topology to Hochschild cohomology.

ABSTRACT

Let $M$ be a 1-connected closed manifold and $LM$ be the space of free loops on $M$. In \cite{C-S} M. Chas and D. Sullivan defined a structure of BV-algebra on the singular homology of $LM$, $H_\ast(LM; \bk)$. When the field of coefficients is of characteristic zero, we prove that there exists a BV-algebra structure on $\hH^\ast(C^\ast (M); C^\ast (M))$ which carries the canonical structure of Gerstenhaber algebra. We construct then an isomorphism of BV-algebras between $\hH^\ast (C^\ast (M); C^\ast (M)) $ and the shifted $ H_{\ast+m} (LM; {\bk})$. We also prove that the Chas-Sullivan product and the BV-operator behave well with the Hodge decomposition of $H_\ast (LM) $.

Motivation & Objective

• To establish a BV-algebra structure on $\mathrm{HH}^\ast(C^\ast(M); C^\ast(M))$ that extends its canonical Gerstenhaber algebra structure.
• To prove an isomorphism of BV-algebras between $H_{\ast+m}(LM)$ and $\mathrm{HH}^\ast(C^\ast(M); C^\ast(M))$ over a field of characteristic zero.
• To show that the Chas-Sullivan product and BV operator are compatible with the Hodge decomposition of $H_{\ast}(LM)$.

Proposed method

• Use a Poincaré duality model $A$ quasi-isomorphic to $C^\ast(M)$, which is finite-dimensional and satisfies Poincaré duality in degree $m$.
• Define a degree $m$ map $\mu_A: A \to A \otimes A$ via duality from the multiplication $\mu$, ensuring compatibility with the differential and $A$-bimodule structure.
• Construct a quasi-isomorphism $f: \mathbf{C}^\ast(\mathfrak{M}_M; \mathfrak{M}_M) \to (\mathfrak{M}_M \otimes \bigwedge \bar{V}, \bar{d})$ using the minimal model of $M$, enabling transfer of algebraic structures.
• Leverage the Hodge decomposition of $H_{\ast}(LM)$ via the filtration $H^{[p]}_{\ast}(LM)$, induced by the cochain complex $\mathcal{C}^\ast(L; UL^\vee_a)$, and relate it to $\mathrm{Tor}^{UL}(\mathbb{K}, \Gamma^p)$.
• Use the Poincaré-Birkhoff-Witt isomorphism $\gamma: \bigwedge L \to UL$ to identify $\Gamma^p = \gamma(\bigwedge^p V)$ and relate $H^{[p]}_{\ast}(LM)$ to $\mathrm{Tor}^{UL}(\mathbb{K}, \Gamma^p)$.
• Prove that the BV operator and product on $H_{\ast+m}(LM)$ correspond under the isomorphism to the canonical BV structure on $\mathrm{HH}^\ast(C^\ast(M); C^\ast(M))$.

Experimental results

Research questions

• RQ1 Does $\mathrm{HH}^\ast(C^\ast(M); C^\ast(M))$ admit a BV-algebra structure extending its Gerstenhaber algebra structure over a field of characteristic zero?
• RQ2 Is there a BV-algebra isomorphism between $H_{\ast+m}(LM)$ and $\mathrm{HH}^\ast(C^\ast(M); C^\ast(M))$ for 1-connected manifolds?
• RQ3 Do the Chas-Sullivan product and BV operator on $H_{\ast}(LM)$ respect the Hodge decomposition induced by the filtration $H^{[p]}_{\ast}(LM)$?

Key findings

• There exists a canonical BV-algebra structure on $\mathrm{HH}^\ast(C^\ast(M); C^\ast(M))$ when the coefficient field has characteristic zero and $M$ is 1-connected.
• The shifted homology $H_{\ast+m}(LM)$ is isomorphic to $\mathrm{HH}^\ast(C^\ast(M); C^\ast(M))$ as BV-algebras.
• The Chas-Sullivan product and BV operator on $H_{\ast}(LM)$ are compatible with the Hodge decomposition $H^{[p]}_{\ast}(LM)$, which corresponds to $\mathrm{Tor}^{UL}(\mathbb{K}, \Gamma^p)$ under duality.
• The isomorphism between $H_{\ast}(LM)$ and $\mathrm{HH}^\ast(C^\ast(M); C^\ast(M))$ is compatible with the Hodge filtration, preserving the BV structure.
• The construction relies on a Poincaré duality model $A$ of $C^\ast(M)$, which allows transfer of the BV structure from $H_{\ast+m}(LM)$ to Hochschild cohomology.
• The result extends previous isomorphisms of Gerstenhaber algebras to the full BV-algebra level, resolving a long-standing question in string topology over characteristic zero fields.