Chas–Sullivan-like Product in String Topology

• Chas–Sullivan-like product is a string-topological operation defined via an intersect–concatenate mechanism that underpins loop and path products on manifolds.
• It extends classical constructions by incorporating umkehr maps, DG/Morse models, Hochschild chains, and equivariant techniques for richer algebraic insights.
• These methods reveal nontrivial algebraic structures linking geometric intersections with homological theories, opening avenues for further research.

Searching arXiv for the cited works on Chas–Sullivan-like products and related string topology constructions. {"query":"(Berest et al., 2016) Chas-Sullivan Lie algebra Hodge filtration derived Poisson brackets (Hingston et al., 2013) path space CP^n RP^n Pontryagin-Chas-Sullivan product (Riegel, 2024) chain-level model Chas-Sullivan products Morse homology differential graded coefficients (Stegemeyer, 2023) String topology on the space of paths with endpoints in a submanifold (Stegemeyer, 27 Mar 2025) Extensions of the loop product and coproduct, the space of antipodal paths and resonances of closed geodesics","max_results":10} Found several relevant arXiv records, including the foundational and recent works on free-loop, equivariant, path-space, and Morse-theoretic extensions of Chas–Sullivan-type operations. “Chas–Sullivan-like product” (Editor's term) denotes a family of string-topological operations modeled on the original Chas–Sullivan loop product: one first imposes an intersection condition on a base manifold, typically through a diagonal or related embedding and its umkehr/Gysin map, and then performs concatenation in a loop or path fiber. In its classical form, for a closed oriented $d$-manifold $M$, the operation

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

is defined using the fiber product $LM\times_M LM$, the diagonal $\Delta:M\to M\times M$, and loop concatenation (Hingston et al., 2017). Subsequent work extends this pattern to $S^1$-equivariant settings, Borel constructions, path spaces with endpoint conditions, DG and $A_\infty$-coefficient models, Novikov completions, Gorenstein spaces, and algebraic models arising from Hochschild and cyclic homology (Kaji et al., 2015, Stegemeyer, 2023, Riegel, 2024, Riegel, 13 Aug 2025).

1. Classical prototype and the intersect–concatenate mechanism

The basic geometric input is the evaluation map at a marked point. For a closed oriented $d$-manifold $M$, one considers

$$LM\times_M LM=\{(\gamma,\sigma)\in LM\times LM\mid \gamma(0)=\sigma(0)\},$$

together with concatenation $M$0. The Chas–Sullivan product is obtained by taking the cross product on $M$1, applying the umkehr map induced by the diagonal $M$2, and then pushing forward by concatenation; the degree shift is $M$3 (Kupper et al., 2022). An equivalent chain-level model uses a tubular neighborhood of the diagonal, the pulled-back Thom class, and a retraction defined by inserting short geodesic “sticks,” avoiding infinite-dimensional tubular-neighborhood arguments (Hingston et al., 2017).

This pattern persists in later generalizations. In the Morse-theoretic framework with DG coefficients, the same operation is described as “intersect on the base, concatenate on the fiber,” with $M$4 viewed as a Hurewicz fibration with fiber $M$5 (Riegel, 2024). In that model the chain-level formula is

$M$6

with a Dold sign inserted to enforce graded commutativity (Riegel, 2024).

The same formal mechanism also underlies the Chas–Sullivan product on compact globally symmetric spaces, where the product is constructed from the pullback of the diagonal and loop concatenation, but is analyzed using explicit Bott–Samelson and Ziller cycles (Kupper et al., 2022). There the product is shown to be highly non-trivial in any rank, and orientation classes of completing manifolds multiply according to iteration of closed geodesics (Kupper et al., 2022).

2. Chain models, DG coefficients, and Hochschild realizations

A major development is the replacement of singular-chain constructions by finite-dimensional algebraic or Morse-theoretic models. In enriched Morse homology with DG or $M$7-module coefficients over $M$8, one forms

$M$9

with differential determined by a Barraud–Cornea twisting cocycle $\mu: H_p(LM)\otimes H_q(LM)\to H_{p+q-d}(LM)$0 satisfying a Maurer–Cartan equation (Riegel, 2024). The resulting chain-level product $\mu: H_p(LM)\otimes H_q(LM)\to H_{p+q-d}(LM)$1 induces the geometric product of Gruher–Salvatore on homology and is associative up to homotopy when the fiberwise multiplication is associative (Riegel, 2024).

A complementary algebraic realization uses Hochschild chains. For a simply connected manifold $\mu: H_p(LM)\otimes H_q(LM)\to H_{p+q-d}(LM)$2, Jones’ cosimplicial model gives a chain homotopy equivalence

$\mu: H_p(LM)\otimes H_q(LM)\to H_{p+q-d}(LM)$3

and Cohen–Jones identify the Chas–Sullivan product with cup product on Hochschild cohomology $\mu: H_p(LM)\otimes H_q(LM)\to H_{p+q-d}(LM)$4 (Maiti, 2019). In that setting, the Gysin map for the diagonal is encoded by the diagonal class

$\mu: H_p(LM)\otimes H_q(LM)\to H_{p+q-d}(LM)$5

in a Poincaré duality CDGA model $\mu: H_p(LM)\otimes H_q(LM)\to H_{p+q-d}(LM)$6 (Maiti, 2019). This produces explicit chain maps modeling degenerate coproducts and the Goresky–Hingston product on relative Hochschild chains (Maiti, 2019).

For simply connected $\mu: H_p(LM)\otimes H_q(LM)\to H_{p+q-d}(LM)$7-Gorenstein spaces, Félix–Thomas string topology replaces geometric Thom data by an Ext-class. If $\mu: H_p(LM)\otimes H_q(LM)\to H_{p+q-d}(LM)$8 is simply connected and $\mu: H_p(LM)\otimes H_q(LM)\to H_{p+q-d}(LM)$9-Gorenstein of formal dimension $LM\times_M LM$0, then the free loop homology $LM\times_M LM$1 carries a non-unital, non-counital Frobenius algebra structure, with loop product and loop coproduct defined through Eilenberg–Moore maps and a generator of

$LM\times_M LM$2

(Naito, 2013). The paper resolves the “up to constant problem,” proving strict associativity, coassociativity, and Frobenius compatibility identities (Naito, 2013).

3. Path-space and relative extensions

The classical loop product has several path-space analogues in which the diagonal condition is imposed on endpoint data rather than loop basepoints. For paths in $LM\times_M LM$3 with endpoints in $LM\times_M LM$4, Hingston–Oancea define

$LM\times_M LM$5

and construct the Pontryagin–Chas–Sullivan product

$LM\times_M LM$6

where $LM\times_M LM$7 is the umkehr map for the matching-endpoint fiber product and $LM\times_M LM$8 is path concatenation (Hingston et al., 2013). In shifted path homology $LM\times_M LM$9, the product has degree $\Delta:M\to M\times M$0, and the paper gives explicit noncommutative ring presentations in terms of generators $\Delta:M\to M\times M$1 or $\Delta:M\to M\times M$2, depending on the parity of $\Delta:M\to M\times M$3 (Hingston et al., 2013).

A general path-space theory for a map $\Delta:M\to M\times M$4 is developed by Stegemeyer. The pulled-back path space

$\Delta:M\to M\times M$5

admits a product

$\Delta:M\to M\times M$6

defined by intersecting along the diagonal $\Delta:M\to M\times M$7 and then concatenating (Stegemeyer, 2023). The product is associative and unital, with unit given by constant paths, and $\Delta:M\to M\times M$8 carries a left module structure over the Chas–Sullivan ring $\Delta:M\to M\times M$9 (Stegemeyer, 2023). If $S^1$0 is null-homotopic, then

$S^1$1

and the product becomes an explicit combination of the intersection pairing on $S^1$2 and the Pontryagin product on $S^1$3 (Stegemeyer, 2023).

This path product was subsequently lifted to a chain-level Morse model with DG coefficients. For a continuous map $S^1$4, the path product

$S^1$5

is defined as

$S^1$6

with a normalization sign $S^1$7, where $S^1$8, $S^1$9 is induced by path concatenation, and $A_\infty$0 is the DG Künneth map (Riegel, 13 Aug 2025). The construction is associative, has a unit, and is shown to agree with Stegemeyer’s singular-homology product under the fibration theorem (Riegel, 13 Aug 2025).

A different relative extension uses a fixed-point-free involution $A_\infty$1 and the path space

$A_\infty$2

In that setting one obtains a pairing

$A_\infty$3

as well as left and right $A_\infty$4-module structures on $A_\infty$5; together they make $A_\infty$6 into a unital associative algebra (Stegemeyer, 27 Mar 2025). For antipodal paths on even-dimensional spheres, the paper computes both the extended loop product and the corresponding extended coproduct explicitly (Stegemeyer, 27 Mar 2025).

4. Equivariant, gauge-theoretic, and Novikov generalizations

An equivariant refinement replaces $A_\infty$7 by the Borel construction $A_\infty$8 for a compact Lie group $A_\infty$9 acting on $d$0. The diagonal of $d$1 does not directly admit a finite-codimension Thom collapse, so the paper factors it through

$d$2

where $d$3 has normal bundle $d$4 and $d$5 has fiber $d$6 with tangent bundle $d$7 (Kaji et al., 2015). Under the orientability hypothesis on both $d$8 and $d$9, one obtains a Borel string product

$M$0

for any multiplicative generalized homology theory $M$1, unifying the Chas–Sullivan product with the Chataur–Menichi product on $M$2 (Kaji et al., 2015). The same paper proves a vanishing theorem for the primary product in a large degree range and constructs a secondary product of degree $M$3 (Kaji et al., 2015).

A spectrum-level generalization arises from a principal $M$4-bundle $M$5. The associated adjoint bundle $M$6 gives the “string topology spectrum”

$M$7

which Gruher–Salvatore showed is a ring spectrum whose homology product is of Chas–Sullivan type (Cohen et al., 2013). In the universal case $M$8 contractible, $M$9, recovering the spectrum-level realization of the classical loop product (Cohen et al., 2013). The same work identifies a gauge-group action

$$LM\times_M LM=\{(\gamma,\sigma)\in LM\times LM\mid \gamma(0)=\sigma(0)\},$$0

and interprets $$LM\times_M LM=\{(\gamma,\sigma)\in LM\times LM\mid \gamma(0)=\sigma(0)\},$$1 as the universal linear approximation to a gauge-theoretic functor (Cohen et al., 2013).

A more recent extension places the same intersect–concatenate paradigm in Morse–Novikov homology. For a closed oriented connected manifold $$LM\times_M LM=\{(\gamma,\sigma)\in LM\times LM\mid \gamma(0)=\sigma(0)\},$$2, a nonzero class $$LM\times_M LM=\{(\gamma,\sigma)\in LM\times LM\mid \gamma(0)=\sigma(0)\},$$3, and a fibration $$LM\times_M LM=\{(\gamma,\sigma)\in LM\times LM\mid \gamma(0)=\sigma(0)\},$$4 with $$LM\times_M LM=\{(\gamma,\sigma)\in LM\times LM\mid \gamma(0)=\sigma(0)\},$$5, one replaces $$LM\times_M LM=\{(\gamma,\sigma)\in LM\times LM\mid \gamma(0)=\sigma(0)\},$$6 by its Novikov completion $$LM\times_M LM=\{(\gamma,\sigma)\in LM\times LM\mid \gamma(0)=\sigma(0)\},$$7 and builds a Morse–Novikov complex with DG coefficients (Riegel, 13 Aug 2025). If the fibration is equipped with a fiberwise multiplication $$LM\times_M LM=\{(\gamma,\sigma)\in LM\times LM\mid \gamma(0)=\sigma(0)\},$$8, the resulting Chas–Sullivan-like product is

$$LM\times_M LM=\{(\gamma,\sigma)\in LM\times LM\mid \gamma(0)=\sigma(0)\},$$9

with associativity, graded commutativity under a commutativity hypothesis on $M$00, and compatibility with a multiplicative spectral sequence (Riegel, 13 Aug 2025). The paper notes that in general there is no unit when $M$01, since classical Morse–Novikov theory gives $M$02 (Riegel, 13 Aug 2025).

5. Algebraic interpretations: Hodge theory, BV reduction, and string brackets

Although the product is the primary focus, several papers place Chas–Sullivan-type operations inside larger algebraic structures. For a simply connected space $M$03 of finite rational type, the Lie model $M$04 yields isomorphisms

$M$05

and the derived Poisson bracket on $M$06 induced by a cyclic pairing on the Koszul dual coalgebra coincides, for a simply connected closed manifold $M$07, with the Chas–Sullivan string bracket on $M$08 (Berest et al., 2016). The same paper constructs Hodge decompositions

$M$09

proves that the derived Poisson bracket preserves the corresponding filtration, and deduces a natural Hodge filtration on the Chas–Sullivan Lie algebra of any simply connected closed manifold (Berest et al., 2016).

A different algebraic reduction appears in the study of the string bracket. Over $M$10, if a closed simply connected manifold is BV-exact, equivalently if the $M$11-action on negative cyclic homology is trivial, then the Chas–Sullivan string bracket on $M$12 is reduced to the loop product followed by the BV operator on loop homology (Kuribayashi et al., 2021). The paper proves that spaces with positive weights are BV-exact and gives parallel results for the dual string cobracket on $M$13 (Kuribayashi et al., 2021). This does not redefine the loop product itself, but it shows that other string-topological operations may collapse to combinations of the loop product and the circle action.

These algebraic models suggest that “Chas–Sullivan-like” behavior is not tied to a single geometric category. In the data above, the same structure appears in universal enveloping algebras, cyclic and Hochschild homology, DG and $M$14-module Morse models, and rational homotopy theory (Berest et al., 2016, Maiti, 2019, Kuribayashi et al., 2021). A plausible implication is that the defining feature is the compatibility of an intersection-type umkehr map with a concatenation-type monoidal operation, rather than the specific choice of loop space.

6. Computations, vanishing phenomena, and open directions

Explicit computations show both richness and rigidity. On compact symmetric spaces, there are many non-nilpotent classes in $M$15 whose powers under the Chas–Sullivan product correspond to iterates of closed geodesics (Kupper et al., 2022). For antipodal path spaces on even-dimensional spheres, the extended algebra $M$16 admits an explicit presentation over $M$17, and the extended coproduct yields resonance theorems for closed geodesics on $M$18 (Stegemeyer, 27 Mar 2025). For $M$19, the Pontryagin–Chas–Sullivan product is explicitly noncommutative and reflects the parity of $M$20 through different ring relations (Hingston et al., 2013).

Vanishing results are equally prominent. The string topology coproduct, often viewed as a counterpart to the Chas–Sullivan product, is substantially harder to compute and is not homotopy invariant in general (Kupper et al., 2024). Intersection multiplicity methods show that the free coproduct vanishes on products $M$21 with $M$22, and that the based coproduct vanishes for total spaces of fiber bundles with sections (Kupper et al., 2024). In higher-rank symmetric spaces, the based coproduct is trivial even though the Chas–Sullivan product remains highly non-trivial (Kupper et al., 2022). These contrasts underscore that product and coproduct behave very differently under geometric constraints.

Several open directions are explicit in the literature. For simply connected closed manifolds, the Hodge filtration on the Chas–Sullivan Lie algebra is proved, but it is conjectured that the bracket actually preserves the Hodge decomposition, not merely the filtration (Berest et al., 2016). For homogeneous spaces such as $M$23 and $M$24, detailed free-loop cohomology calculations are available, but the Chas–Sullivan product itself is not yet computed explicitly; the spectral-sequence data are presented as the input for such a computation (Burfitt, 2017). In the Morse–Novikov setting, the product is constructed, but BV and Gerstenhaber refinements are not developed (Riegel, 13 Aug 2025). Recent Morse-theoretic work on path products and DG coefficients likewise suggests further extensions to coproducts and higher string operations (Riegel, 13 Aug 2025).

Taken together, these results portray the Chas–Sullivan-like product as a broad structural motif in string topology: an operation of degree determined by a codimension, built by umkehr along a diagonal or diagonal-like embedding, followed by concatenation in a loop or path direction. Its incarnations range from free loops to endpoint-constrained paths, from equivariant Borel homology to Novikov completions, and from geometric intersections to Hochschild- and Ext-theoretic models (Kaji et al., 2015, Stegemeyer, 2023, Riegel, 2024, Naito, 2013).