Loop space homology of elliptic spaces

Abstract

In this thesis, we use the theory of minimal Sullivan models in rational homotopy theory to study the partial computation of the Lie bracket structure of the string homology on a formal elliptic space. In the process, we show the total space of the unit sphere tangent bundleS2m−1 → Ep→ Gk,n(C) over complex Grassmannian manifolds Gk,n(C) for 2 ≤ k ≤ n/2, where m = k(n − k) is not formal. This is done by exhibiting a non trivial Massey triple product. On the other hand, let φ :(∧V,d) → (B,d) be a surjective morphism between com mutative differential graded algebras, where V is finite dimensional, and consider (B,d) a module over ∧V via the mapping φ. We show that the Hochschild cohomology HH∗ (∧V;B) can be computed in terms of the graded vector space of positive φ-derivations. Given a Koszul Sullivan extension (∧V,d) f ↣ (∧V ⊗ ∧W,d) = (C,d), we show that if (∧V,d) is an elliptic 2-stage Postnikov tower Sullivan algebra, and if the natural homo morphism of the differential graded algebras (C,d) → (∧W,d¯) is surjective in homology, then the natural graded linear map HH∗ (f) : HH∗ (∧V;∧V) → HH∗ (∧V;C), induced in Hochschild cohomology by the inclusion (∧V,d) f ↣ (C,d), is injective. In particular, if X is an elliptic 2-stage Postnikov tower, and (∧V,d) is the minimal Sullivan model of X, then HH∗ (f) : H∗(X S 1 ;Q) → HH∗ (∧V;C) is injective, where X S 1 is the space of free loops on X, and H∗(X S 1 ;Q) is the loop space homology.