On the invariance of the string topology coproduct

We show that the Goresky-Hingston coproduct in string topology is invariant under homotopy equivalences that satisfy an additional assumption, either stated in terms the restriction to the “fake diagonal”, or in terms of boundedness for the homotopy. Introduction For a closed oriented manifold M , the Chas-Sullivan product [4] is a product on the homology of its free loop space ΛM = Maps(S,M) of the form ∧ : Hp(ΛM)⊗Hq(ΛM) −→ Hp+q−n(ΛM), where H∗(−) = H∗(−;Z) denotes singular homology with integral coefficients and n = dimM . Goresky and Hingston defined in [10] a “dual” product in cohomology relative to the constant loops ~ : H(ΛM,M)⊗H(ΛM,M)→ Hp+q+n−1(ΛM,M). The associated homology coproduct ∨ : Hp(ΛM,M) −→ Hp−n+1(ΛM × ΛM,M × ΛM ∪ ΛM ×M). was defined by Sullivan [23]. The idea of the Chas-Sullivan product is to intersect the chains of basepoints of two chains of loops and concatenate the loops at the common basepoints. The homology coproduct looks instead for self-intersections at the basepoint within a single chain of loops and cuts. Both structures are represented geometrically by the same figure, either thought of as two loops intersecting at their basepoint, or as a single loop having a self-intersection at its basepoint. For the coproduct, it is essential to look for self-intersections with the basepoint at all times t along the loop, including t = 0 and t = 1 (where all loops tautologically have a self-intersection!)—Tamanoi showed that the coproduct that only looks for self-intersections at t = 12 is trivial [24]. In contrast, H ∗(ΛSn, S) for an odd sphere S is generated by just 4 classes as an algebra with the cohomology product ~ [10, 15.3]. The domain [0, 1] of the time variable t is the reason why the coproduct naturally is an operation in relative homology: it has nontrivial boundary terms coming from the boundary of the interval. This operation can be associated to a 1-parameter family of surfaces in the harmonic compactification of the moduli space of Riemann surfaces, making it a so-called compactified operation. This coproduct detects intersection multiplicity of homology classes in the loop space, the largest number k such that any representing chain of a given homology class in H∗(ΛM) necessarily has a loop with a k-fold self-intersection at its basepoint. (See [14, Thm 3.10].) Date: September 9, 2021. 1As we work with homology with integral coefficients, we do not in general have a Künneth isomorphism, which is why the coproduct in integral homology has target H∗(ΛM ×ΛM,M ×ΛM ∪ΛM ×M) rather than H∗(ΛM,M)⊗H∗(ΛM,M). 1 2 NANCY HINGSTON AND NATHALIE WAHL While the above products and coproduct are all induced by chain maps (see eg., [14, Sec 1]), we will only consider in the present paper the induced maps in homology/cohomology. It has been shown by several authors that the Chas-Sullivan product is homotopy invariant, in the sense that a homotopy equivalence f : M1 → M2 induces an isomorphism Λf : H∗(ΛM1)→ H∗(ΛM2) of algebras with respect to the Chas-Sullivan product (see [7, Thm 1],[11, Prop 23],[8, Thm 3.7]). In contrast, a coproduct of the same flavour was used in [5, Sec 2.1] to define the differential in string homology, a knot invariant that detects the unknot. Also the paper [19] found indications of non-homotopy invariance in the S–equivariant version of the coproduct. And indeed, Naef showed recently through a computation with lens spaces that the coproduct is not homotopy invariant [18]. (See also Appendix B for an overview of Naef’s example.) Note though that, over the rational and for simply-connected manifolds, Rivera-Wang showed that the Goresky-Hingston coproduct is homotopy invariant using a Hochschild homology model [22]. We give here two sufficient conditions for homotopy equivalences to respect the coproduct, one formulated in terms of the boundedness of the homotopy equivalence (condition B), and one formulated in terms of the “fake diagonal” (condition D): A map f : M1 → M2 is a homotopy equivalence if one can find a homotopy inverse g together with homotopies g ◦ f 'h1 idM1 and f ◦ g 'h2 idM2 . We say that f is a bounded homotopy equivalence with respect to a metric on M2 if one can find g, h1, h2 such that h2 stays bounded and h2 ◦ (f × I) stays close to f ◦ h1 (see Definition 3.9). Our first condition on f is the following: Condition B: The map f is a bounded homotopy equivalence with respect to a Riemannian metric on M2. For the other sufficient condition, let ∆i ∈Mi ×Mi denote the diagonal. We call ∆f1 = {(m,m′) ∈M1 ×M1 | f(m) = f(m′)} the fake diagonal. Note that ∆1 ⊂ ∆f1 = (f × f)(∆2). Our alternative condition on f is: Condition D: The inclusion ∆1 ↪→ ∆f1 is a weak homotopy equivalence. Both conditions are trivially satisfied if f is a diffeomorphism. Condition D is also satisfied by elementary collapses, i.e. maps contracting a cell in M1, see Example 2.2. The two conditions will in addition require a “goodness assumption” on f , that we call ∆-transversality (see Definition 2.3). We expect that any f is homotopic to a map that is ∆-transverse (see Conjecture 2.4). For f not ∆-transverse, conditions D can instead be replaced by more technical versions of the conditions, involving neighborhoods of the fake diagonal, see condition D’ in Section 2.1. Conditions B and D both have the same purpose: ensuring that a certain map j0 : B ε2 X2 → B2 1,2 is a homology isomorphism, for certain spaces of “half-small loops” B2 X2 and B ε2 1,2 defined in Section 2. We show in Proposition 2.5 that, under the ∆–transversality assumption, this map j0 is a weak homotopy equivalence if and only if condition D holds. We expect that condition B is a strictly stronger condition. Theorem A. Let f : M1 →M2 be a homotopy equivalence between closed oriented manifolds, homotopic to a ∆–transverse map satisfying either condition B or condition D, or to a map satisfying condition D’. Then the map induced by Λf : ΛM1 → ΛM2 in homology respects the coproduct, up to a sign given by the degree of f , that is the following diagram commute: