String Topology Coproduct

• String topology coproduct is a relative homology operation on free loop spaces that cuts loops at self-intersections using evaluation maps and Thom class pullbacks.
• It combines geometric methods with algebraic techniques like Hochschild and coHochschild models to capture manifold invariants and detect torsion phenomena.
• The operation is highly sensitive to global topology and intersection multiplicity, vanishing under certain homotopy conditions while distinguishing subtle manifold structures.

The string topology coproduct is the degree $1-n$ operation on the relative homology of the free loop space $LM=\operatorname{Map}(S^1,M)$ of a closed oriented $n$-manifold $M$ that cuts loops at self-intersections detected by the diagonal in $M\times M$. In its standard Goresky–Hingston form it is defined on $H_*(LM,M)$, is dual to a cohomology product of degree $n-1$, and sits alongside the Chas–Sullivan product as one of the two major string topology operations (Naef et al., 2022).

1. Geometric definition and relative formulation

For a closed oriented $n$-manifold $M$, the Goresky–Hingston coproduct is the relative homology operation

$$\Delta_{\mathrm{GH}}\colon H_i(LM,M;R)\longrightarrow H_{i+1-n}(LM\times LM,\;LM\times M\cup M\times LM;R),$$

of degree $LM=\operatorname{Map}(S^1,M)$0 (Stegemeyer, 2021). Its construction uses the evaluation map

$LM=\operatorname{Map}(S^1,M)$1

from $LM=\operatorname{Map}(S^1,M)$2 to $LM=\operatorname{Map}(S^1,M)$3, the pullback of a Thom class of the diagonal, a retraction from a neighborhood of “near self-intersections” onto the exact self-intersection locus

$LM=\operatorname{Map}(S^1,M)$4

and a cutting map that splits a loop at the marked time $LM=\operatorname{Map}(S^1,M)$5 into two loops (Stegemeyer, 2021). In the notation used in the Lie-group specialization, the defining formula is

$LM=\operatorname{Map}(S^1,M)$6

The relative nature is essential. The interval parameter $LM=\operatorname{Map}(S^1,M)$7 introduces boundary terms at $LM=\operatorname{Map}(S^1,M)$8 and $LM=\operatorname{Map}(S^1,M)$9, and the passage to $n$0 removes contributions from constant loops and “constant ears” (Naef et al., 2022). A fixed-time cut coproduct

$n$1

can be defined on $n$2, but it is trivial except possibly in low degrees; the nontrivial string topology operation is the relative Goresky–Hingston coproduct built from the full $n$3-family of cut times (Kupper et al., 2024).

There is also a based version. For a basepoint $n$4, the same construction yields

$n$5

and the inclusion $n$6 is compatible with the free coproduct via a commutative square on relative homology (Stegemeyer, 2021). This based coproduct is often the technically simpler object and, in several families of examples, controls the free coproduct.

2. Configuration-space and Hochschild models

A geometric reformulation replaces tubular neighborhoods by the compactified configuration space of two points. For a closed manifold $n$7, the Fulton–MacPherson–Axelrod–Singer compactification $n$8 of $n$9 has boundary $M$0 and fits into a homotopy pushout

$M$1

which encodes the diagonal embedding without choosing a tubular neighborhood (Naef et al., 2019). Evaluating a loop at two marked points and cutting at those points produces a chain-level zigzag whose homology-level output is the Goresky–Hingston coproduct. In the $M$2-framed case $M$3, the relative Thom class can be replaced by a fiberwise volume form on $M$4, giving an absolute version on $M$5 (Naef et al., 2019).

In simply connected cases, the coproduct admits explicit Hochschild descriptions. If $M$6 is a Poincaré duality cdga model of $M$7 with diagonal class $M$8, the cohomology product dual to the Goresky–Hingston coproduct is modeled on reduced Hochschild chains by

$M$9

with the usual Koszul signs (Naef et al., 2019). The same paper constructs an $M\times M$0-structure on cyclic chains, identifies negative cyclic homology with $M\times M$1-equivariant loop homology, and shows that the induced bracket and cobracket on equivariant homology reproduce the string topology Lie bialgebra structure (Naef et al., 2019).

A more recent algebraic model uses the coHochschild complex of a coalgebra of chains equipped with a local pairing on simplicial chains. In that framework, the Chas–Sullivan product and the Goresky–Hingston coproduct are defined directly on the coHochschild complex using local higher homotopies controlling compatibility with the diagonal approximation coproduct; for closed oriented smooth manifolds, the algebraic operations coincide, up to chain homotopy, with the geometric ones (Rivera et al., 21 Aug 2025). The local nature of the construction allows arguments based on the method of acyclic models (Rivera et al., 21 Aug 2025).

3. Algebraic structure and relation to other string topology operations

At chain level, Hingston–Wahl introduced a signed version of the coproduct whose homology-level form is better behaved. On $M\times M$2, the signed coproduct is the twisted suspension of a graded coassociative, cocommutative coproduct, and the dual cohomology operation on $M\times M$3 is the twisted desuspension of a graded associative, commutative product (Kupper et al., 2024). In particular, the cohomology-side Goresky–Hingston product is not unital, and its signs differ from the Chas–Sullivan product by the extra suspension coming from the interval parameter.

The coproduct is not merely a coalgebraic analogue of the Chas–Sullivan product. For the lifted absolute coproduct $M\times M$4 constructed by extension from relative chains, Hingston–Wahl showed that the composition with the Chas–Sullivan product vanishes: $M\times M$5 on homology (Hingston et al., 2017). The same work emphasizes a geometric interpretation: nonvanishing of the $M\times M$6-th iterate of the coproduct forces the existence of a loop with a $M\times M$7-fold self-intersection in every representative of the class, and on spheres and projective spaces this criterion is sharp (Hingston et al., 2017).

Passing to $M\times M$8-equivariant homology, the coproduct yields the string cobracket. If $M\times M$9 is the homotopy quotient, composition with the Gysin maps defines degree-$H_*(LM,M)$0 operations

$H_*(LM,M)$1

recovering the Chas–Sullivan/Turaev Lie bialgebra structure (Naef et al., 2019). This places the Goresky–Hingston coproduct in the same formal package as the BV operator and the Chas–Sullivan product, even though the coproduct itself is naturally defined only in reduced or relative form.

4. Computations in Lie groups, symmetric spaces, projective spaces, and surfaces

For a connected Lie group $H_*(LM,M)$2, the free loop space splits topologically: $H_*(LM,M)$3 Using Bott’s theorem and a Künneth decomposition, the coproduct on $H_*(LM,M)$4 splits exactly as

$H_*(LM,M)$5

where $H_*(LM,M)$6 is the diagonal and $H_*(LM,M)$7 is the based coproduct on $H_*(LM,M)$8 (Stegemeyer, 2021). This has strong consequences: if $H_*(LM,M)$9 is even, then $n-1$0 for degree reasons, hence $n-1$1; if $n-1$2 is compact, simply connected, and of rank at least $n-1$3, Bott–Samelson cycles imply that every class has basepoint intersection multiplicity $n-1$4, so the coproduct again vanishes (Stegemeyer, 2021). The nontrivial low-rank odd case persists: for $n-1$5, the splitting theorem applies but the triviality theorems do not, and the coproduct is nontrivial (Stegemeyer, 2021).

Complex and quaternionic projective spaces admit explicit generators from Ziller’s completing manifolds. If $n-1$6 and $n-1$7 denote the standard rational generators in $n-1$8, then

$n-1$9

and

$n$0

for $n$1 or $n$2 (Stegemeyer, 2022). Dualizing gives a complete description of the Goresky–Hingston product on $n$3, including finitely generated presentations and non-nilpotent generators (Stegemeyer, 2022).

For compact globally symmetric spaces of rank at least $n$4, Bott–Samelson and Ziller cycles lead to extensive vanishing. The based string topology coproduct is trivial for compact simply connected symmetric spaces of rank $n$5, and the free coproduct vanishes on large natural subspaces, while the Chas–Sullivan product remains highly nontrivial and detects iteration of closed geodesics (Kupper et al., 2022). This produces a sharp contrast between product and coproduct in higher rank.

On closed oriented surfaces of genus $n$6, the coproduct admits a complete combinatorial description in terms of cyclic words in generators of $n$7. The induced string cobracket on $n$8 is the negative of the Turaev cobracket: $n$9 under the canonical identification of equivariant degree-zero homology with free homotopy classes (Hartenstein et al., 7 Oct 2025). In this case the coproduct is fully computable by an explicit algorithm on cyclic words, and the higher-genus surface case becomes one of the rare settings with a complete closed-form answer (Hartenstein et al., 7 Oct 2025).

5. Homotopy invariance, torsion, and manifold sensitivity

The coproduct is not homotopy invariant in general. Over $M$0 and for simply connected manifolds, the geometric coproduct agrees with an algebraic Hochschild model and is homotopy invariant under quasi-isomorphisms of the underlying Frobenius model (Naef et al., 2022). Over $M$1, however, lens-space calculations show that the coproduct detects data beyond ordinary homotopy type (Naef et al., 2022).

For the lens spaces $M$2, the reduced coproduct can be written explicitly in terms of Reidemeister torsion. With $M$3, one has

$M$4

and the reduced coproducts on $M$5 and $M$6 are different (Naef, 2021). In particular, the coproduct coalgebras are non-isomorphic, so the operation distinguishes homotopy equivalent manifolds (Naef, 2021).

Naef–Safronov refined this phenomenon by proving a transformation formula under orientation-preserving homotopy equivalences. The defect in functoriality is expressed by the Dennis trace of the Whitehead torsion,

$M$7

and the formula implies that the loop coproduct is invariant under simple homotopy equivalences (Naef et al., 2024). The same work identifies the coproduct as a secondary operation in a $M$8-dimensional TQFT and shows that framed configuration spaces of at most two points determine the Dennis trace of the simple homotopy type (Naef et al., 2024). The upshot is that the coproduct is sensitive not just to the homotopy type of $M$9, but to simple-homotopy and torsion-theoretic data encoded near the diagonal.

6. Vanishing criteria and extensions beyond the closed-manifold case

A systematic vanishing criterion is provided by intersection multiplicity. For a relative class $$\Delta_{\mathrm{GH}}\colon H_i(LM,M;R)\longrightarrow H_{i+1-n}(LM\times LM,\;LM\times M\cup M\times LM;R),$$0, if $$\Delta_{\mathrm{GH}}\colon H_i(LM,M;R)\longrightarrow H_{i+1-n}(LM\times LM,\;LM\times M\cup M\times LM;R),$$1, then the $$\Delta_{\mathrm{GH}}\colon H_i(LM,M;R)\longrightarrow H_{i+1-n}(LM\times LM,\;LM\times M\cup M\times LM;R),$$2-fold iterated coproduct vanishes: $$\Delta_{\mathrm{GH}}\colon H_i(LM,M;R)\longrightarrow H_{i+1-n}(LM\times LM,\;LM\times M\cup M\times LM;R),$$3 On spheres and projective spaces, the converse holds: vanishing of $$\Delta_{\mathrm{GH}}\colon H_i(LM,M;R)\longrightarrow H_{i+1-n}(LM\times LM,\;LM\times M\cup M\times LM;R),$$4 is equivalent to the bound $$\Delta_{\mathrm{GH}}\colon H_i(LM,M;R)\longrightarrow H_{i+1-n}(LM\times LM,\;LM\times M\cup M\times LM;R),$$5 (Kupper et al., 2024). In the based setting, $$\Delta_{\mathrm{GH}}\colon H_i(LM,M;R)\longrightarrow H_{i+1-n}(LM\times LM,\;LM\times M\cup M\times LM;R),$$6 already forces the based coproduct to vanish (Kupper et al., 2024).

This criterion yields large vanishing families. If $$\Delta_{\mathrm{GH}}\colon H_i(LM,M;R)\longrightarrow H_{i+1-n}(LM\times LM,\;LM\times M\cup M\times LM;R),$$7 is a product of closed manifolds with $$\Delta_{\mathrm{GH}}\colon H_i(LM,M;R)\longrightarrow H_{i+1-n}(LM\times LM,\;LM\times M\cup M\times LM;R),$$8, then every class in $$\Delta_{\mathrm{GH}}\colon H_i(LM,M;R)\longrightarrow H_{i+1-n}(LM\times LM,\;LM\times M\cup M\times LM;R),$$9 has $LM=\operatorname{Map}(S^1,M)$00, so the free coproduct vanishes (Kupper et al., 2024). If $LM=\operatorname{Map}(S^1,M)$01 is a smooth fiber bundle with section and both base and fiber are positive-dimensional closed manifolds, then every class in $LM=\operatorname{Map}(S^1,M)$02 has $LM=\operatorname{Map}(S^1,M)$03, hence the based coproduct on $LM=\operatorname{Map}(S^1,M)$04 is trivial (Kupper et al., 2024). Tori, higher-rank compact Lie groups, and higher-rank symmetric spaces fit into this general vanishing pattern (Kupper et al., 2024).

Beyond closed oriented manifolds, string topology coproducts extend to several broader frameworks. For simply connected $LM=\operatorname{Map}(S^1,M)$05-Gorenstein spaces, the shifted homology $LM=\operatorname{Map}(S^1,M)$06 carries a non-unital, non-counital Frobenius algebra structure with loop product and loop coproduct defined via Sullivan models and Eilenberg–Moore maps; odd-generated and even-generated minimal models exhibit sharply different triviality behavior (Naito, 2013). In derived string topology, the Eilenberg–Moore spectral sequence for a simply connected Gorenstein space admits a multiplication and a comultiplication compatible with the loop product and loop coproduct of the target (Kuribayashi et al., 2012). A different extension replaces free loops by the path space

$LM=\operatorname{Map}(S^1,M)$07

attached to an involution $LM=\operatorname{Map}(S^1,M)$08; in this setting one obtains extended loop products and coproducts that recover the classical operations when $LM=\operatorname{Map}(S^1,M)$09, and the antipodal involution on spheres yields complete computations and applications to resonances of closed geodesics on real projective space (Stegemeyer, 27 Mar 2025).

Taken together, these developments show that the string topology coproduct is simultaneously geometric, algebraic, and highly sensitive to global topology. It is often trivial on large classes of spaces, but when nontrivial it detects self-intersection geometry, configuration-space data, torsion phenomena, and, in special families, explicit representation-theoretic or combinatorial structures.