An Algebraic Chain Model of String Topology

A chain complex model for the free loop space of a connected, closed and oriented manifold is presented, and on its homology, the Gerstenhaber and Batalin-Vilkovisky algebra structures are defined and identified with the string topology structures. The gravity algebra on the equivariant homology of the free loop space is also modeled. The construction includes non simply-connected case, and therefore gives an algebraic and chain level model of Chas-Sullivan’s String Topology.

💡 Research Summary

The paper presents a comprehensive chain‑level model for the free loop space (LM=\operatorname{Map}(S^{1},M)) of a connected, closed, oriented manifold (M), and shows how the three fundamental algebraic structures of string topology—Gerstenhaber, Batalin‑Vilkovisky (BV), and gravity algebras—arise naturally on the homology of this model. The construction works over the rational numbers and extends to non‑simply‑connected manifolds, thereby providing a fully algebraic and chain‑level realization of Chas‑Sullivan string topology.

1. DG Open Frobenius‑like Algebra.
The authors start by introducing a pair of differential graded (DG) objects associated to (M):

• (A(M)), the DG algebra of Whitney polynomial differential forms on a smooth cubulation of (M). These forms have rational polynomial coefficients on each cube, are compatible under face restrictions, and carry the usual wedge product (\wedge) and exterior differential (d).
• (C(M)=\operatorname{Hom}(A(M),\mathbb{Q})), the DG coalgebra of currents (the linear dual of (A(M))). The coproduct (\Delta) on (C(M)) is defined as the dual of the wedge product on (A(M)).

The inclusion (\iota:A(M)\hookrightarrow C(M)) given by Poincaré duality is a quasi‑isomorphism of DG modules. Moreover, the Frobenius compatibility condition
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