Abstract. Let G be a finite group or a compact connected Lie group and let BG be its classifying space. Let LBG := map(S 1 , BG) be the free loop space of BG i.e. the space of continuous maps from the circle S 1 to BG. The purpose of this paper is to study the singular homology H * (LBG) of this loop space. We prove that when taken with coefficients in a field the homology of LBG is a homological conformal field theory. As a byproduct of our main theorem, we get a Batalin-Vilkovisky algebra structure on the cohomology H * (LBG). We also prove an algebraic version of this result by showing that the Hochschild cohomology HH

Let X be a pseudomanifold. In this text, we use a simplicial blow-up to define a cochain complex whose cohomology with coefficients in a field, is isomorphic to the intersection cohomology of X, introduced by M. Goresky and R. MacPherson.We do it simplicially in the setting of a filtered version of face sets, also called simplicial sets without degeneracies, in the sense of C.P. Rourke and B.J. Sanderson. We define perverse local systems over filtered face sets and intersection cohomology with coefficients in a perverse local system. In particular, as announced above when X is a pseudomanifold, we get a perverse local system of cochains quasi-isomorphic to the intersection cochains of Goresky and MacPherson, over a field. We show also that these two complexes of cochains are quasi-isomorphic to a filtered version of Sullivan's differential forms over the field Q. In a second step, we use these forms to extend Sullivan's presentation of rational homotopy type to intersection cohomology.For that, we construct a functor from the category of filtered face sets to a category of perverse commutative differential graded Q-algebras (CDGA's) due to Hovey. We establish also the existence and unicity of a positively graded, minimal model of some perverse CDGA's, including the perverse forms over a filtered face set and their intersection cohomology. Finally, we prove the topological invariance of the minimal model of a PLpseudomanifold whose regular part is connected, and this theory creates new topological invariants. This point of view brings a definition of formality in the intersection setting and examples are given. In particular, we show that any nodal hypersurface in CP(4), is intersection-formal.

Topological invariance of the intersection homology of a pseudomanifold without codimension one strata, proven by Goresky and MacPherson, is one of the main features of this homology. This property is true for codimension-dependent perversities with some growth conditions, verifying p(1) = p(2) = 0. King reproves this invariance by associating an intrinsic pseudomanifold X * to any pseudomanifold X. His proof consists of an isomorphism between the associated intersection homologies H p * (X) ∼ = H p * (X * ) for any perversity p with the same growth conditions verifying p(1) ≥ 0.In this work, we prove a certain topological invariance within the framework of strata-dependent perversities, p, which corresponds to the classical topological invariance if p is a GM-perversity. We also extend it to the tame intersection homology, a variation of the intersection homology, particularly suited for "large" perversities, if there is no singular strata on X becoming regular in X * . In particular, under the above conditions, the intersection homology and the tame intersection homology are invariant under a refinement of the stratification. Definition 1.3. A stratified space is a filtered space such that any pair of strata, S andDefinition 1.4. Let X be a stratified space. The depth of X is the greater integer m for which it exists a chain of strata, S 0 ≺ S 1 ≺ · · · ≺ S m . We denote depth X = m.Example 1.5. If X is stratified, each construction of Example 1.2 is a stratified space.Definition 1.6. A stratified map, f : X → Y , is a continuous map between stratified spaces such that, for each stratum S ∈ S X , there exists a unique stratumObserve that a continuous map f : X → Y is stratified if, and only if, the pull-back of a stratum S ′ ∈ S Y is empty or a union f −1 (S ′ ) = ⊔ i∈I S i , with codim S ′ ≤ codim S i for each i ∈ I. Therefore, a stratified map sends a regular stratum in a regular one but the image of a singular stratum can be included in a regular one. Example 1.7. Let X be a stratified space. The canonical projection, pr : M × X → X, the maps ι t : X →cX with x → [x, t], ι m : X → M × X with x → (m, x) and the canonical injection of an open subset U ֒→ X, are stratified for the structures described in Example 1.2. Let us recall some properties of stratified maps from [4, Section A.2]. Proposition 1.8 ([4, Proposition A.23]). A stratified map, f : X → Y , induces the order preserving map (S X , ) → (S Y , ), defined by S → S f .Let us introduce the notion of homotopy between stratified maps. Here, the product X × [0, 1] is endowed with the product filtration. Definition 1.9. Two stratified maps f, g : X → Y are homotopic if there exists a stratified map, ϕ : X × [0, 1] → Y , such that ϕ(−, 0) = f and ϕ(−, 1) = g. Homotopy is an equivalence relation and produces the notion of homotopy equivalence between stratified spaces.The following notion of locally cone-like stratified space has been introduced by Siebenman, [20]. Definition 1.10. A CS set of dimension n is a filtered space, ∅ ⊂ X 0 ⊆ X 1 ⊆ · · · ⊆ X n−2 ...

In this article, we introduce Adem-Cartan operads and prove that the cohomology of any algebra over such an operad is an unstable level algebra over the extended Steenrod algebra. Moreover, we prove that this cohomology is endowed with secondary cohomology operations.

No abstract