A Homotopy Theoretic Realization of String Topology

Let M be a closed, oriented manifold of dimension d. Let LM be the space of smooth loops in M . In [2] Chas and Sullivan defined a product on the homology H∗(LM) of degree −d. They then investigated other structure that this product induces, including a Lie algebra structure on H∗(LM), and an induced product on the S 1 equivariant homology, H S ∗ (LM). These algebraic structures, as well as others, came under the general heading of the “String topology” of M . In this paper we will describe a realization of the Chas Sullivan loop product in terms of a ring spectrum structure on the Thom spectrum of a certain virtual bundle over the loop space. We show that this ring spectrum structure extends to an operad action of the the “cactus operad”, originally defined by Voronov, which is equivalent to operad of framed disks in R. We then describe a cosimplicial model of this spectrum and, by applying the singular cochain functor to this cosimplicial spectrum we show that this ring structure can be interpreted as the cup product in the Hochschild cohomology, HH(C(M);C(M)). Introduction Let M be a closed, oriented d dimensional manifold, and let LM = C∞(S1,M) be the space of smooth loops in M . In [2] Chas and Sullivan described an intersection product on the homology, H∗(LM), having total degree −d, ◦ : Hq(LM)⊗Hr(LM) → Hq+r−d(LM). In this paper we show that this product is realized by a geometric structure, not on the loop space itself, but on the Thom spectrum of a certain bundle over LM . We describe this structure both homotopy theoretically and simplicially, and in so doing describe the relationship of the Chas Sullivan product to the cup product in Hochshild cohomology. We now make these statements more precise. Consider the standard parameterization of the circle by the unit interval, exp : [0, 1] → S1 defined by exp(t) = e2πit. With respect to this parameterization we can regard a loop γ ∈ LM as a map γ : [0, 1] → M with γ(0) = γ(1). Consider the evaluation map Date: February 26, 2008. The first author was partially supported by a grant from the NSF . 1 2 R.L. COHEN AND J.D.S JONES ev : LM → M γ → γ(1). Now let ι : M → RN+d be a fixed embedding of M into codimension N Euclidean space. Let ν → M be the N dimensional normal bundle. Let Th(ν ) be the Thom space of this bundle. Recall the famous result of Atiyah [1] that Th(ν ) is Spanier Whitehead dual to M+. Said more precisely, let M −TM be the spectrum given by desuspending this Thom space, M = ΣTh(ν ). Then if M+ denotes M with a disjoint basepoint, there are maps of spectra S → M+ ∧M −TM and M+ ∧M −TM → S that establish M as the S − dual of M+. Said another way, these maps induce an equivalence with the function spectrum M ≃ Map(M+, S 0). In particular in homology we have isomorphisms H(M+) ∼= H−q(M −TM ) H(M ) ∼= Hq(M+) for all q ∈ Z. These duality isomorphism are induced by the compositions H(M ) τ −−−→ ∼= H−q+d(M) ρ −−−→ ∼= Hq(M) where τ is the Thom isomorphism, and ρ is Poincare duality. Notice by duality, the diagonal map ∆ : M → M ×M induces a map of spectra ∆ : M ∧M → M that makes M into a ring spectrum with unit. The unit S0 → M is the map dual to the projection M+ → S 0. Now let Th(ev(ν )) be the Thom space of the pull back bundle ev(ν ) → LM . Define the spectrum LM = ΣTh(ev(ν )) The goal of this paper is to define and study a product structure on the spectrum LM which among other properties makes the evaluation map ev : LM → M a map of A HOMOTOPY THEORETIC REALIZATION OF STRING TOPOLOGY 3 ring spectra. Here, by abuse of notation, ev is referring the map of Thom spectra induced by the evaluation map ev : LM → M . We will prove the following theorem. Theorem 1. The spectrum LM is a homotopy commutative ring spectrum with unit, whose multiplication μ : LM ∧ LM −→ LM satisfies the following properties. 1. The evaluation map ev : LM → M is a map of ring spectra. 2. There is a map of ring spectra ρ : LM → Σ(ΩM+) where the target is the suspension spectrum of the based loop space with a disjoint basepoint. Its ring structure is induced by the usual product on the based loop space. In homology the map ρ∗ is given by the composition ρ∗ : Hq(LM −TM ) τ −−−→ ∼= Hq+dLM ι −−−→ Hq(ΩM) where like above, τ is the Thom isomorphism, and the map ι takes a (q+ d) cycle in LM and intersects it with the based loop space viewed as a codimension d submanifold. 3. The ring structure is compatible with the Chas Sullivan homology product in the sense that the following diagram commutes: Hq(LM −TM )⊗Hr(LM −TM ) −−−→ Hq+r(LM −TM ∧ LM ) ρ∗ −−−→ Hq+r(LM −TM ) τ 