Chain-Level String Topology

• Chain-level string topology is the study of operations on chain complexes modeling the free-loop space of a closed manifold, integrating algebraic and geometric methods.
• It employs algebraic models like based-loop chains and Hochschild complexes to reconstruct classical operations such as the Chas–Sullivan product and BV structure.
• Properadic, cyclic, and IBL∞ frameworks organize higher-genus and equivariant operations, unifying diverse chain-level approaches in modern topology.

Chain-level string topology is the study of string-topological operations before passage to homology, on explicit chain complexes that model the free loop space $LM$ of a closed oriented manifold $M$, or on algebraic substitutes such as Hochschild chains and cochains. In this setting, the Chas–Sullivan loop product, the circle-action operator, coproducts, brackets, and higher operations are realized as chain maps, operadic actions, properadic operations, or $IBL_\infty$-structures. A central theme is that the homology-level Batalin–Vilkovisky structure on $\mathbb H_*(LM):=H_{*+d}(LM)$ can be reconstructed from the based loop space, from geometric loop-chain models, or from cyclic and Hochschild complexes, with these viewpoints linked by Poincaré duality, Atiyah duality, and Jones-type iterated integral maps (Malm, 2011, Irie, 2014).

1. Algebraic foundations in based-loop chains

A foundational formulation replaces chains on the free loop space by the differential graded Hopf algebra $A=C_*\Omega M$, where $\Omega M$ is the based loop space. For a closed, connected, oriented smooth $d$-manifold $M$, the string-topology BV structure on $\mathbb H_*(LM)$ is generated by the Chas–Sullivan product

$\circ:H_p(LM)\otimes H_q(LM)\to H_{p+q-d}(LM)$

and the degree-$M$0 operator $M$1 induced by the rotation action of $M$2 on $M$3. Malm showed that this entire BV algebra can be reconstructed from Hochschild theory of $M$4 (Malm, 2011).

The algebraic bridge has three components. First, the Goodwillie–Burghelea–Fiedorowicz identification gives

$M$5

and intertwines Connes’ operator $M$6 with loop rotation: $M$7 Second, derived Poincaré duality for $M$8-modules identifies

$M$9

for any $IBL_\infty$0-module $IBL_\infty$1, by capping with a fundamental class in $IBL_\infty$2. Third, Hochschild chains and cochains with coefficients in $IBL_\infty$3 are identified with $IBL_\infty$4 and $IBL_\infty$5 over $IBL_\infty$6 using the adjoint module $IBL_\infty$7. For $IBL_\infty$8, this yields

$IBL_\infty$9

Combining these ingredients gives an additive isomorphism

$\mathbb H_*(LM):=H_{*+d}(LM)$0

and hence

$\mathbb H_*(LM):=H_{*+d}(LM)$1

Pulling Connes’ operator back along $\mathbb H_*(LM):=H_{*+d}(LM)$2 defines

$\mathbb H_*(LM):=H_{*+d}(LM)$3

The triple $\mathbb H_*(LM):=H_{*+d}(LM)$4 is then a BV algebra; its BV-derived bracket agrees with the usual Gerstenhaber bracket on Hochschild cohomology, and under $\mathbb H_*(LM):=H_{*+d}(LM)$5 the cup product corresponds to the loop product while $\mathbb H_*(LM):=H_{*+d}(LM)$6 corresponds to $\mathbb H_*(LM):=H_{*+d}(LM)$7. In this formulation, all string-topology operations are encoded in Hochschild cohomology of the based-loop chain algebra. A notable feature is that no simply-connectedness assumption is required; the constructions are formulated for $\mathbb H_*(LM):=H_{*+d}(LM)$8-oriented Poincaré duality spaces and in particular for closed oriented manifolds.

2. Geometric chain models on the free loop space

A second line of work realizes string topology directly on geometric chain complexes for $\mathbb H_*(LM):=H_{*+d}(LM)$9. Irie introduced de Rham chains $A=C_*\Omega M$0 for differentiable spaces $A=C_*\Omega M$1, defined from plots $A=C_*\Omega M$2 together with differential forms on $A=C_*\Omega M$3, modulo relations generated by integration along fibers. These complexes admit functorial cross products and fiber products, and for manifolds and loop spaces satisfy

$A=C_*\Omega M$4

Using Moore loops with marked points, Irie defined differentiable spaces $A=C_*\Omega M$5 and a nonsymmetric cyclic dg operad

$A=C_*\Omega M$6

Ward’s cyclic Deligne theory then provides an action of a chain model of the framed little disks operad on the associated total complex $A=C_*\Omega M$7, and

$A=C_*\Omega M$8

as BV algebras, with the induced multiplication and circle operator recovering the Chas–Sullivan product and BV operator (Irie, 2014).

A complementary geometric compactification was developed using spaces $A=C_*\Omega M$9 of string diagrams, which are compact CW complexes of dimension

$\Omega M$0

For each type $\Omega M$1, Poirier–Rounds constructed a chain map

$\Omega M$2

defined by evaluation at chord endpoints, pullback of a Thom class of the multidiagonal, cap product, and geodesic filling of the chords. After quotienting by slide equivalence, the induced homology operations recover the Cohen–Godin positive-boundary TQFT (Poirier et al., 2011).

A further refinement gives explicit integral chain-level constructions of the loop product, the Goresky–Hingston coproduct, and a non-relative cohomology product. In that approach, all operations are written as compositions of singular-chain maps built from evaluation maps, cap products with a pulled-back Thom class, retractions defined by adding short geodesic “sticks,” and cutting or concatenation maps. The lifted coproduct

$\Omega M$3

and the extended cohomology product

$\Omega M$4

are compatible with the length filtration, and the identity

$\Omega M$5

holds on homology (Hingston et al., 2017). These geometric constructions avoid infinite-dimensional tubular neighborhoods by pulling all Thom data back from finite-dimensional neighborhoods of the diagonal in $\Omega M$6.

3. Properads, ribbon graphs, and universal operations

Chain-level string topology also admits a properadic organization. Merkulov constructed the chain gravity properad $\Omega M$7 from ribbon graphs with white and black vertices, obtained by twisting a ribbon-graph properad by a degree-shifted Lie structure. This properad acts canonically on cyclic Hochschild complexes of cyclic $\Omega M$8-algebras, and in particular on reduced cyclic Hochschild complexes of Poincaré duality algebras. Quotienting by the properadic ideal generated by graphs with black vertices of valency at least $\Omega M$9 or boundaries consisting only of black vertices yields the chain string topology properad

$d$0

For a Poincaré duality algebra $d$1 of degree $d$2, the properad $d$3 acts canonically on

$d$4

and its cohomology properad $d$5 acts on the reduced $d$6-equivariant homology $d$7 of the free loop space of any connected, simply connected closed $d$8-manifold (Merkulov, 2022).

This properadic framework organizes several previously distinct structures in one object. The cohomology properad $d$9 sits under the properad of involutive Lie bialgebras, so its action recovers the Chas–Sullivan involutive Lie bialgebra on $M$0. It also sits under the properad of homotopy involutive Lie bialgebras, exhibiting higher homotopy operations beyond the primary bracket and cobracket. In addition, the gravity operad $M$1 injects into $M$2, so classes in compactly supported cohomology of moduli spaces $M$3 yield universal string-topology operations. In this sense, moduli of curves, ribbon-graph models, and chain-level cyclic Hochschild operations are assembled into a single properadic calculus.

4. $M$4-equivariant, cyclic, and $M$5 formulations

A cocyclic approach makes the $M$6-equivariant aspect explicit. For a space $M$7 with circle action, the cocyclic spaces $M$8 yield mixed complexes $M$9 and $\mathbb H_*(LM)$0, and their cyclic homology computes $\mathbb H_*(LM)$1-equivariant homology. In the de Rham-chain setting, the same construction applies to Irie’s loop operad $\mathbb H_*(LM)$2, producing the cyclic-invariant complex

$\mathbb H_*(LM)$3

whose homology is identified with negative $\mathbb H_*(LM)$4-equivariant homology: $\mathbb H_*(LM)$5 Ward’s operad $\mathbb H_*(LM)$6 acts on $\mathbb H_*(LM)$7, so this complex carries a homotopy gravity algebra whose homology recovers the string-topology gravity algebra (Wang, 2022).

A parallel and more field-theoretic formulation uses the dual cyclic bar complex

$\mathbb H_*(LM)$8

of a harmonic model $\mathbb H_*(LM)$9. On this complex there is a canonical dIBL structure, and perturbative Chern–Simons theory produces a Maurer–Cartan element by configuration-space integrals over ribbon graphs. This yields an $\circ:H_p(LM)\otimes H_q(LM)\to H_{p+q-d}(LM)$0-structure that models equivariant string topology on cyclic Hochschild cochains of de Rham cohomology (Hajek, 2020). A later existence and uniqueness theorem states that there is a Maurer–Cartan element

$\circ:H_p(LM)\otimes H_q(LM)\to H_{p+q-d}(LM)$1

for the canonical dIBL structure on

$\circ:H_p(LM)\otimes H_q(LM)\to H_{p+q-d}(LM)$2

unique up to $\circ:H_p(LM)\otimes H_q(LM)\to H_{p+q-d}(LM)$3-gauge equivalence, such that the twisted homology of $\circ:H_p(LM)\otimes H_q(LM)\to H_{p+q-d}(LM)$4 is canonically isomorphic to Connes’ cyclic cohomology $\circ:H_p(LM)\otimes H_q(LM)\to H_{p+q-d}(LM)$5. For simply connected $\circ:H_p(LM)\otimes H_q(LM)\to H_{p+q-d}(LM)$6, this realizes chain-level $\circ:H_p(LM)\otimes H_q(LM)\to H_{p+q-d}(LM)$7-equivariant string topology (Cieliebak et al., 2023).

The analytic completion of this picture uses configuration spaces, propagators on the real blow-up $\circ:H_p(LM)\otimes H_q(LM)\to H_{p+q-d}(LM)$8, Chen’s iterated integrals, and homotopy transfer. In that framework, the map

$\circ:H_p(LM)\otimes H_q(LM)\to H_{p+q-d}(LM)$9

from $M$00 to the homology of a twisted dIBL complex intertwines the string bracket and cobracket with the dIBL bracket and cobracket. This identifies the involutive Lie bialgebra on equivariant loop homology with the one obtained from homotopy transfer of the de Rham DGA and perturbative Chern–Simons theory (Cieliebak et al., 4 Jul 2026).

5. Coproducts, self-intersections, and algebraic analogues

The loop coproduct and its relatives are among the most delicate chain-level operations. The explicit chain models for the lifted coproduct $M$01 show that iterating the coproduct detects self-intersections of loops: if $M$02, then every chain representative of $M$03 contains a loop with a $M$04-fold self-intersection. For spheres and projective spaces, this criterion is sharp: $M$05 exactly when $M$06 can be represented by loops with at most $M$07-fold self-intersections (Hingston et al., 2017).

Hochschild chain models make these operations algebraically explicit. For a Poincaré duality CDGA $M$08 modeling a simply connected closed manifold, the degenerate coproducts are represented by concrete maps

$M$09

and a chain homotopy

$M$10

satisfying $M$11. Restricting $M$12 to the reduced Hochschild complex yields a chain model of the Goresky–Hingston product. In the same framework, the degenerate string coproduct is shown to be almost trivial, and nonnilpotent local level classes for the Chas–Sullivan and Goresky–Hingston products detect closed geodesics with optimal index growth rates (Maiti, 2019).

A noncommutative algebraic analogue replaces $M$13 by Hochschild chains $M$14 of a smooth $M$15-category equipped with a pre-Calabi–Yau structure $M$16 and a trivialization of the chain-level Chern character $M$17 of the diagonal bimodule. From a trivialization $M$18 of $M$19, one constructs

$M$20

which is the algebraic loop coproduct. It satisfies a Sullivan-type infinitesimal bialgebra relation with the algebraic loop product $M$21,

$M$22

and under balanced and symmetry conditions induces cocommutative and coassociative structures on reduced Hochschild homology (Rivera et al., 2023). This suggests that the Goresky–Hingston coproduct has a natural extension from manifold models to smooth $M$23-categories and formal punctured neighborhoods of infinity.

6. Examples, obstructions, and comparison phenomena

Concrete examples show that chain-level refinements are not optional. For $M$24 over $M$25, Menichi had shown that the string-topology BV algebra on

$M$26

is not isomorphic, as a BV algebra, to the BV structure on

$M$27

obtained using only ordinary Poincaré duality on cohomology. Poirier–Tradler resolved this discrepancy by replacing strict cohomological duality with a homotopy-coherent Poincaré duality structure on cochains, described by a homotopy inner product $M$28. The transferred higher-homotopy duality on $M$29 then yields a BV operator on Hochschild cohomology that matches the string-topology BV operator (Poirier et al., 2023).

This example isolates a general issue. Strict algebraic models may correctly recover the Gerstenhaber structure while still missing the correct BV operator. By contrast, the based-loop model through $M$30 encodes the BV structure by transporting Connes’ $M$31-operator across derived Poincaré duality, and does so without a simply-connectedness hypothesis (Malm, 2011). The discrepancy on $M$32 therefore should not be read as a failure of Hochschild methods; rather, it indicates that the relevant duality data must be imposed at chain level and with higher homotopies.

A broader comparison emerges from the literature. Geometric loop-chain models retain evaluation maps, geodesic retractions, and filtrations by length or energy; based-loop and Hochschild models give strict algebraic control through Hopf algebras, bar constructions, and cyclic operators; properadic and $M$33 models organize higher-genus and equivariant operations uniformly. A plausible implication is that “chain-level string topology” is not a single complex but a web of quasi-equivalent realizations, each emphasizing a different structural feature: BV and Gerstenhaber structures, coproduct and self-intersection phenomena, properadic operations from moduli spaces, or $M$34-equivariant Lie bialgebra structures.