Derived string topology and the Eilenberg-Moore spectral sequence

Abstract

Let M be any simply-connected Gorenstein space over any field. Félix and Thomas have extended to simply-connected Gorenstein spaces, the loop (co)products of Chas and Sullivan on the homology of the free loop space H * (LM ). We describe these loop (co)products in terms of the torsion and extension functors by developing string topology in appropriate derived categories. As a consequence, we show that the Eilenberg-Moore spectral sequence converging to the loop homology of a Gorenstein space admits a multiplication and a comultiplication with shifted degree which are compatible with the loop product and the loop coproduct of its target, respectively.We also define a generalized cup product on the Hochschild cohomology HH * (A, A ∨ ) of a commutative Gorenstein algebra A and show that over Q, HH * (A P L (M ), A P L (M ) ∨ ) is isomorphic as algebras to H * (LM ). Thus, when M is a Poincaré duality space, we recover the isomorphism of algebras H * (LM ; Q) ∼ = HH * (A P L (M ), A P L (M )) of Félix and Thomas.

Behavior of the Eilenberg–Moore spectral sequence in derived string topology

“…Proof. The result [12,Theorem 8.6 (1) and (2)] enable us to obtain the commutative squares up to homotopy…”

“…We recall the product m i on the loop homology H * (L αi M ) = H * +di (L αi M ) defined by [12,Lemma 8.6] for the sign. Then, we see that…”

“…Therefore (1) can be proved easily. But in order to prove part (2), we need to interpret (1) in term of differential torsion product described in [12,Theorem 2.3]; see the proof of Propositions 4.1. and 4.3).…”

“…Let {F p } p≥0 be the filtration of the loop homolog H * (L N M ) which the spectral sequence {E * , * r , d r } provides. Then we see that (F p ) n = 0 for p > dim N − n; see [12,Remark 6.1]. This yields that I · a = a for any a in (F p ) n with p = dim N − n. Suppose that I · Q = Q for any Q ∈ (F >s ) n .…”

Produit de Chas–Sullivan et actions d'un groupe de Lie connexe

“…Dans un premier temps (section 2), nous établirons une description du dual du loop produit pour un espace de Gorenstein de dimension formelle n en termes de modèles de Sullivan (cf. [5,2,6]). Ensuite (section 3), nous donnerons le modèle minimal de Sullivan de l'espace E Γ lorsque M est un espace homogène G/H via une action de Γ sur le groupe de Lie compacte connexe G et finalement (section 3), nous indiquerons un exemple avec une action de Γ = S 1 sur G = U (n + 1) dépendant d'un paramètre λ = 0, 1.…”