String topology on Gorenstein spaces

Abstract

The purpose of this paper is to describe a general and simple setting for defining (g, p + q)-string operations on a Poincaré duality space and more generally on a Gorenstein space. Gorenstein spaces include Poincaré duality spaces as well as classifying spaces or homotopy quotients of connected Lie groups. Our presentation implies directly the homotopy invariance of each (g, p + q)-string operation as well as it leads to explicit computations. IntroductionShriek maps play a central role in string topology and its generalizations. Following Dold, various presentations have been given [1,3,5,10,30]. Usually shriek maps are defined in (co)homology for maps f : N → M from a closed oriented n-manifold to a closed oriented m-manifold. Here we will consider shriek maps at the cochain level and in a more general setting.More precisely we work in the category of (left or right) differential graded modules over a differential graded lk-algebra (R, d) (lk is a fixed field), that we call for sake of simplicity the category of (R, d)-modules. Its associated derived category [20] is obtained by formally inverting quasi-isomorphisms, i.e. the homomorphisms of (R, d)-modules that induce isomorphisms in homology and hereafter denoted by in the diagrams. In the derived category the vector space of Y. Félix

Coproducts in brane topology

“…Generalizing these constructions, Félix and Thomas [FT09] defined the loop product and coproduct in the case M is a Gorenstein space. A Gorenstein space is a generalization of a manifold in the point of view of Poincaré duality, including the classifying space of a connected Lie group and the Borel construction of a connected oriented closed manifold and a connected Lie group.…”

“…Let K be a field of characteristic zero. For example, the loop coproduct is trivial for a manifold with the Euler characteristic zero [Tam10, Corollary 3.2], the composition of the loop coproduct followed by the loop product is trivial for any manifold [Tam10, Theorem A], and the loop product over K is trivial for the classifying space of a connected Lie group [FT09,Theorem 14]. A space with the nontrivial composition of loop coproduct and product is not found.…”

“…In a usual sense, the base map c is not an embedding of finite codimension. But using the algebraic method of Félix and Thomas [FT09], we can consider this map as an embedding of codimensionm = dim ΩM , which is the dimension as a K-Gorenstein space and is finite when π * (M ) ⊗ K is of finite dimension. Hence, under this assumption, we have the shriek map c !…”

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.…”

A note on the Chas-Sullivan product

“…According to their abstract, they describe "a realization of the ChasSullivan product in terms of a ring spectrum structure on the Thom spectrum of a certain virtual bundle over the loop space", a difficult technique indeed. Recently in [8], Y. Félix and J.-C. Thomas have put String Topology into a broad homotopy theoretical setting; they prove that the operations in string topology are preserved by homotopy equivalence, at least in the 1-connected case. On the contrary, in the present note we propose a finite dimensional approach, very close to the spirit of [4], based on transversality arguments.…”